Theorem selvply1rhmlemb 33706

Description: Lemma for selvply1rhm 33712. (Contributed by Thierry Arnoux, 4-May-2026.)

Hypotheses

Ref	Expression
selvply1rhmlema.1	⊢ 𝐵 = (Base‘𝑃)
selvply1rhmlema.2	⊢ 𝑃 = ({𝑋} mPoly 𝑅)
selvply1rhmlema.3	⊢ · = (.r‘𝑃)
selvply1rhmlema.4	⊢ × = (.r‘𝑄)
selvply1rhmlema.5	⊢ 𝑄 = (Poly1‘𝑅)
selvply1rhmlema.6	⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})))
selvply1rhmlema.7	⊢ (𝜑 → 𝑋 ∈ 𝑉)
selvply1rhmlema.8	⊢ (𝜑 → 𝑅 ∈ Ring)
selvply1rhmlema.9	⊢ (𝜑 → 𝐹 ∈ 𝐵)
selvply1rhmlemb.10	⊢ (𝜑 → 𝐺 ∈ 𝐵)

Assertion

Ref	Expression
selvply1rhmlemb	⊢ (𝜑 → (𝑀‘(𝐹 · 𝐺)) = ((𝑀‘𝐹) × (𝑀‘𝐺)))

Distinct variable groups:   · ,𝑓   𝐵,𝑓   𝑓,𝐹,𝑛   𝑅,𝑛   𝑓,𝑋,𝑛   𝜑,𝑓,𝑛   × ,𝑓   · ,𝑛   𝑓,𝐺,𝑛   𝑓,𝑀,𝑛

Allowed substitution hints:   𝐵(𝑛)   𝑃(𝑓,𝑛)   𝑄(𝑓,𝑛)   𝑅(𝑓)   × (𝑛)   𝑉(𝑓,𝑛)

Proof of Theorem selvply1rhmlemb

Dummy variables 𝑚 ℎ 𝑖 𝑗 𝑥 𝑔 𝑙 𝑘 𝑜 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		selvply1rhmlema.6	. 2 ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})))
2		fveq1 6829	. . . 4 ⊢ (𝑓 = (𝐹 · 𝐺) → (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}) = ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩}))
3	2	mpteq2dv 5169	. . 3 ⊢ (𝑓 = (𝐹 · 𝐺) → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩})))
4		selvply1rhmlema.2	. . . . . . . 8 ⊢ 𝑃 = ({𝑋} mPoly 𝑅)
5		selvply1rhmlema.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑃)
6		eqid 2736	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
7		selvply1rhmlema.3	. . . . . . . 8 ⊢ · = (.r‘𝑃)
8		eqid 2736	. . . . . . . . 9 ⊢ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} = {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}
9	8	psrbasfsupp 33698	. . . . . . . 8 ⊢ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} = {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ (◡𝑔 “ ℕ) ∈ Fin}
10		selvply1rhmlema.9	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
11		selvply1rhmlemb.10	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
12	4, 5, 6, 7, 9, 10, 11	mplmul 21988	. . . . . . 7 ⊢ (𝜑 → (𝐹 · 𝐺) = (𝑚 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ↦ (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗)))))))
13	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝐹 · 𝐺) = (𝑚 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ↦ (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗)))))))
14		breq2 5079	. . . . . . . . . 10 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑙 ∘r ≤ 𝑚 ↔ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
15	14	rabbidv 3395	. . . . . . . . 9 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} = {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}})
16		fvoveq1 7382	. . . . . . . . . 10 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝐺‘(𝑚 ∘f − 𝑗)) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)))
17	16	oveq2d 7375	. . . . . . . . 9 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗))) = ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))
18	15, 17	mpteq12dv 5162	. . . . . . . 8 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗)))) = (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)))))
19	18	oveq2d 7375	. . . . . . 7 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))))
20		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
21		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2736	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
23		fveq2 6830	. . . . . . . . . 10 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → (𝐹‘𝑗) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
24		oveq2 7367	. . . . . . . . . . 11 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) = ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))
25	24	fveq2d 6834	. . . . . . . . . 10 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
26	23, 25	oveq12d 7377	. . . . . . . . 9 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))) = ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))
27		selvply1rhmlema.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
28	27	ringcmnd 20259	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CMnd)
29	28	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑅 ∈ CMnd)
30		eqid 2736	. . . . . . . . . . 11 ⊢ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} = {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}
31		ovexd 7394	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℕ0 ↑m {𝑋}) ∈ V)
32	8, 31	rabexd 5271	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∈ V)
33	30, 32	rabexd 5271	. . . . . . . . . 10 ⊢ (𝜑 → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ∈ V)
34	33	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ∈ V)
35		fvexd 6845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (0g‘𝑅) ∈ V)
36	32	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∈ V)
37		ssrab2 4014	. . . . . . . . . . 11 ⊢ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}
38	37	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
39	4, 21, 5, 9, 11	mplelf 21975	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
40	39	ad2antrr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝐺:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
41		breq1 5078	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑔 finSupp 0 ↔ {⟨𝑋, (𝑛‘∅)⟩} finSupp 0))
42		nn0ex 12437	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ∈ V
43	42	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ℕ0 ∈ V)
44		snex 5371	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ∈ V
45	44	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑋} ∈ V)
46		selvply1rhmlema.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
47	46	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑋 ∈ 𝑉)
48		1oex 8408	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ V
49	48	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 1o ∈ V)
50		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛 ∈ (ℕ0 ↑m 1o))
51	49, 43, 50	elmaprd 32775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛:1o⟶ℕ0)
52		0lt1o 8432	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ 1o
53	52	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ∅ ∈ 1o)
54	51, 53	ffvelcdmd 7029	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛‘∅) ∈ ℕ0)
55	47, 54	fsnd 6814	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩}:{𝑋}⟶ℕ0)
56	43, 45, 55	elmapdd 8781	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩} ∈ (ℕ0 ↑m {𝑋}))
57		snfi 8983	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ∈ Fin
58	57	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑋} ∈ Fin)
59		c0ex 11132	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
60	59	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 0 ∈ V)
61	55, 58, 60	fdmfifsupp 9281	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩} finSupp 0)
62	41, 56, 61	elrabd 3634	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
63	62	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨𝑋, (𝑛‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
64	44	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {𝑋} ∈ V)
65	42	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ℕ0 ∈ V)
66		ssrab2 4014	. . . . . . . . . . . . . . . . 17 ⊢ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ⊆ (ℕ0 ↑m {𝑋})
67	37, 66	sstri 3927	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ (ℕ0 ↑m {𝑋})
68	67	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ (ℕ0 ↑m {𝑋}))
69	68	sselda 3918	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∈ (ℕ0 ↑m {𝑋}))
70	64, 65, 69	elmaprd 32775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗:{𝑋}⟶ℕ0)
71		breq1 5078	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑗 → (𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ↔ 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
72		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}})
73	71, 72	elrabrd 32589	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩})
74	9	psrbagcon 21903	. . . . . . . . . . . . 13 ⊢ (({⟨𝑋, (𝑛‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∧ 𝑗:{𝑋}⟶ℕ0 ∧ 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∧ ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
75	63, 70, 73, 74	syl3anc 1375	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∧ ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
76	75	simpld 495	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
77	40, 76	ffvelcdmd 7029	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)) ∈ (Base‘𝑅))
78	4, 21, 5, 9, 10	mplelf 21975	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
79	78	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝐹:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
80	4, 5, 22, 10	mplelsfi 21972	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
81	80	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝐹 finSupp (0g‘𝑅))
82	27	ad2antrr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
83		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
84	21, 6, 22, 82, 83	ringlzd 20270	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑥) = (0g‘𝑅))
85	35, 35, 36, 38, 77, 79, 81, 84	fisuppov1 32778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)))) finSupp (0g‘𝑅))
86		ssidd 3941	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (Base‘𝑅) ⊆ (Base‘𝑅))
87	27	ad2antrr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑅 ∈ Ring)
88	78	ad2antrr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝐹:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
89	38	sselda 3918	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
90	88, 89	ffvelcdmd 7029	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝐹‘𝑗) ∈ (Base‘𝑅))
91	21, 6, 87, 90, 77	ringcld 20235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))) ∈ (Base‘𝑅))
92		breq1 5078	. . . . . . . . . 10 ⊢ (𝑙 = {⟨𝑋, (𝑖‘∅)⟩} → (𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ↔ {⟨𝑋, (𝑖‘∅)⟩} ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
93		breq1 5078	. . . . . . . . . . 11 ⊢ (𝑔 = {⟨𝑋, (𝑖‘∅)⟩} → (𝑔 finSupp 0 ↔ {⟨𝑋, (𝑖‘∅)⟩} finSupp 0))
94	42	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ℕ0 ∈ V)
95	44	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {𝑋} ∈ V)
96	47	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑋 ∈ 𝑉)
97	48	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 1o ∈ V)
98		ssrab2 4014	. . . . . . . . . . . . . . . . 17 ⊢ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ⊆ (ℕ0 ↑m 1o)
99	98	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ⊆ (ℕ0 ↑m 1o))
100	99	sselda 3918	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∈ (ℕ0 ↑m 1o))
101	97, 94, 100	elmaprd 32775	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖:1o⟶ℕ0)
102	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∅ ∈ 1o)
103	101, 102	ffvelcdmd 7029	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ∈ ℕ0)
104	96, 103	fsnd 6814	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩}:{𝑋}⟶ℕ0)
105	94, 95, 104	elmapdd 8781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∈ (ℕ0 ↑m {𝑋}))
106	57	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {𝑋} ∈ Fin)
107	59	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 0 ∈ V)
108	104, 106, 107	fdmfifsupp 9281	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} finSupp 0)
109	93, 105, 108	elrabd 3634	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
110		simplr 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑛 ∈ (ℕ0 ↑m 1o))
111		breq1 5078	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ∘r ≤ 𝑛 ↔ 𝑖 ∘r ≤ 𝑛))
112		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛})
113	111, 112	elrabrd 32589	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∘r ≤ 𝑛)
114		elmapfn 8805	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℕ0 ↑m 1o) → 𝑖 Fn 1o)
115	114	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 𝑖 Fn 1o)
116		elmapfn 8805	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → 𝑛 Fn 1o)
117	116	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 𝑛 Fn 1o)
118	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 1o ∈ V)
119		inidm 4158	. . . . . . . . . . . . . 14 ⊢ (1o ∩ 1o) = 1o
120		eqidd 2737	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) ∧ ∅ ∈ 1o) → (𝑖‘∅) = (𝑖‘∅))
121		eqidd 2737	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) ∧ ∅ ∈ 1o) → (𝑛‘∅) = (𝑛‘∅))
122	115, 117, 118, 118, 119, 120, 121	ofrval 7635	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∘r ≤ 𝑛 ∧ ∅ ∈ 1o) → (𝑖‘∅) ≤ (𝑛‘∅))
123	110, 100, 113, 102, 122	syl211anc 1380	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ≤ (𝑛‘∅))
124	123	ralrimivw 3132	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∀𝑥 ∈ {𝑋} (𝑖‘∅) ≤ (𝑛‘∅))
125	104	ffnd 6659	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} Fn {𝑋})
126	55	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑛‘∅)⟩}:{𝑋}⟶ℕ0)
127	126	ffnd 6659	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
128		inidm 4158	. . . . . . . . . . . 12 ⊢ ({𝑋} ∩ {𝑋}) = {𝑋}
129		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → 𝑥 ∈ {𝑋})
130	129	elsnd 4576	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → 𝑥 = 𝑋)
131	130	fveq2d 6834	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑥) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋))
132		fvsng 7127	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑖‘∅) ∈ ℕ0) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
133	96, 103, 132	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
134	133	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
135	131, 134	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑥) = (𝑖‘∅))
136	130	fveq2d 6834	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑥) = ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋))
137	54	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛‘∅) ∈ ℕ0)
138		fvsng 7127	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑛‘∅) ∈ ℕ0) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
139	96, 137, 138	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
140	139	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
141	136, 140	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑥) = (𝑛‘∅))
142	125, 127, 95, 95, 128, 135, 141	ofrfval 7633	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑖‘∅)⟩} ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ↔ ∀𝑥 ∈ {𝑋} (𝑖‘∅) ≤ (𝑛‘∅)))
143	124, 142	mpbird 258	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩})
144	92, 109, 143	elrabd 3634	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}})
145		breq1 5078	. . . . . . . . . . 11 ⊢ (𝑘 = {⟨∅, (𝑗‘𝑋)⟩} → (𝑘 ∘r ≤ 𝑛 ↔ {⟨∅, (𝑗‘𝑋)⟩} ∘r ≤ 𝑛))
146	48	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 1o ∈ V)
147		df1o2 8405	. . . . . . . . . . . . . . 15 ⊢ 1o = {∅}
148	147	eqcomi 2745	. . . . . . . . . . . . . 14 ⊢ {∅} = 1o
149	148	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {∅} = 1o)
150		0ex 5232	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
151	150	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∅ ∈ V)
152		snidg 4595	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
153	46, 152	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
154	153	ad2antrr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑋 ∈ {𝑋})
155	70, 154	ffvelcdmd 7029	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝑗‘𝑋) ∈ ℕ0)
156	151, 155	fsnd 6814	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩}:{∅}⟶ℕ0)
157	149, 156	feq2dd 6644	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩}:1o⟶ℕ0)
158	65, 146, 157	elmapdd 8781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} ∈ (ℕ0 ↑m 1o))
159		simplr 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑛 ∈ (ℕ0 ↑m 1o))
160	47	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑋 ∈ 𝑉)
161	159, 160	jca 512	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉))
162		elmapfn 8805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℕ0 ↑m {𝑋}) → 𝑗 Fn {𝑋})
163	162	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) → 𝑗 Fn {𝑋})
164		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
165		elmapi 8789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → 𝑛:1o⟶ℕ0)
166	52	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → ∅ ∈ 1o)
167	165, 166	ffvelcdmd 7029	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → (𝑛‘∅) ∈ ℕ0)
168	167	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → (𝑛‘∅) ∈ ℕ0)
169	164, 168	fsnd 6814	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → {⟨𝑋, (𝑛‘∅)⟩}:{𝑋}⟶ℕ0)
170	169	ffnd 6659	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
171	170	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
172	44	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) → {𝑋} ∈ V)
173		eqidd 2737	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) ∧ 𝑋 ∈ {𝑋}) → (𝑗‘𝑋) = (𝑗‘𝑋))
174	164, 168, 138	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
175	174	ad2antlr 729	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) ∧ 𝑋 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
176	163, 171, 172, 172, 128, 173, 175	ofrval 7635	. . . . . . . . . . . . . . 15 ⊢ (((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) ∧ 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ∧ 𝑋 ∈ {𝑋}) → (𝑗‘𝑋) ≤ (𝑛‘∅))
177	69, 161, 73, 154, 176	syl211anc 1380	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝑗‘𝑋) ≤ (𝑛‘∅))
178		fveq2 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 = ∅ → (𝑛‘𝑜) = (𝑛‘∅))
179	178	breq2d 5087	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = ∅ → ((𝑗‘𝑋) ≤ (𝑛‘𝑜) ↔ (𝑗‘𝑋) ≤ (𝑛‘∅)))
180	150, 179	ralsn 4616	. . . . . . . . . . . . . 14 ⊢ (∀𝑜 ∈ {∅} (𝑗‘𝑋) ≤ (𝑛‘𝑜) ↔ (𝑗‘𝑋) ≤ (𝑛‘∅))
181	177, 180	sylibr 235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∀𝑜 ∈ {∅} (𝑗‘𝑋) ≤ (𝑛‘𝑜))
182	147	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 1o = {∅})
183	181, 182	raleqtrrdv 3298	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∀𝑜 ∈ 1o (𝑗‘𝑋) ≤ (𝑛‘𝑜))
184	157	ffnd 6659	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} Fn 1o)
185	116	ad2antlr 729	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑛 Fn 1o)
186		elsni 4575	. . . . . . . . . . . . . . . . 17 ⊢ (𝑜 ∈ {∅} → 𝑜 = ∅)
187	186, 147	eleq2s 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ 1o → 𝑜 = ∅)
188	187	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → 𝑜 = ∅)
189	188	fveq2d 6834	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → ({⟨∅, (𝑗‘𝑋)⟩}‘𝑜) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅))
190	155	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → (𝑗‘𝑋) ∈ ℕ0)
191		fvsng 7127	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ V ∧ (𝑗‘𝑋) ∈ ℕ0) → ({⟨∅, (𝑗‘𝑋)⟩}‘∅) = (𝑗‘𝑋))
192	150, 190, 191	sylancr 589	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → ({⟨∅, (𝑗‘𝑋)⟩}‘∅) = (𝑗‘𝑋))
193	189, 192	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → ({⟨∅, (𝑗‘𝑋)⟩}‘𝑜) = (𝑗‘𝑋))
194		eqidd 2737	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → (𝑛‘𝑜) = (𝑛‘𝑜))
195	184, 185, 146, 146, 119, 193, 194	ofrfval 7633	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ({⟨∅, (𝑗‘𝑋)⟩} ∘r ≤ 𝑛 ↔ ∀𝑜 ∈ 1o (𝑗‘𝑋) ≤ (𝑛‘𝑜)))
196	183, 195	mpbird 258	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} ∘r ≤ 𝑛)
197	145, 158, 196	elrabd 3634	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛})
198		eqcom 2743	. . . . . . . . . . . . 13 ⊢ ((𝑗‘𝑋) = (𝑖‘∅) ↔ (𝑖‘∅) = (𝑗‘𝑋))
199	198	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑗‘𝑋) = (𝑖‘∅) ↔ (𝑖‘∅) = (𝑗‘𝑋)))
200	133	adantlr 717	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
201	200	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑗‘𝑋) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) ↔ (𝑗‘𝑋) = (𝑖‘∅)))
202	155	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑗‘𝑋) ∈ ℕ0)
203	150, 202, 191	sylancr 589	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨∅, (𝑗‘𝑋)⟩}‘∅) = (𝑗‘𝑋))
204	203	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑖‘∅) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅) ↔ (𝑖‘∅) = (𝑗‘𝑋)))
205	199, 201, 204	3bitr4d 312	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑗‘𝑋) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) ↔ (𝑖‘∅) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅)))
206	160	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑋 ∈ 𝑉)
207		eqid 2736	. . . . . . . . . . . 12 ⊢ {𝑋} = {𝑋}
208	70	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑗:{𝑋}⟶ℕ0)
209	208	ffnd 6659	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑗 Fn {𝑋})
210	125	adantlr 717	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} Fn {𝑋})
211	206, 207, 209, 210	fsneq 6979	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} ↔ (𝑗‘𝑋) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋)))
212	150	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∅ ∈ V)
213	101	adantlr 717	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖:1o⟶ℕ0)
214	213	ffnd 6659	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 Fn 1o)
215	184	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨∅, (𝑗‘𝑋)⟩} Fn 1o)
216	212, 147, 214, 215	fsneq 6979	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖 = {⟨∅, (𝑗‘𝑋)⟩} ↔ (𝑖‘∅) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅)))
217	205, 211, 216	3bitr4d 312	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} ↔ 𝑖 = {⟨∅, (𝑗‘𝑋)⟩}))
218	197, 217	reu6dv 32563	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∃!𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}𝑗 = {⟨𝑋, (𝑖‘∅)⟩})
219	20, 21, 22, 26, 29, 34, 85, 86, 91, 144, 218	gsummptfsf1o 33144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))))
220	98	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ⊆ (ℕ0 ↑m 1o))
221	220	sselda 3918	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∈ (ℕ0 ↑m 1o))
222		fveq1 6829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (𝑛‘∅) = (𝑖‘∅))
223	222	opeq2d 4814	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑖 → ⟨𝑋, (𝑛‘∅)⟩ = ⟨𝑋, (𝑖‘∅)⟩)
224	223	sneqd 4570	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑖 → {⟨𝑋, (𝑛‘∅)⟩} = {⟨𝑋, (𝑖‘∅)⟩})
225	224	fveq2d 6834	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝐹‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
226		fveq1 6829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝐹 → (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐹‘{⟨𝑋, (𝑛‘∅)⟩}))
227	226	mpteq2dv 5169	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝐹 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})))
228		ovexd 7394	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℕ0 ↑m 1o) ∈ V)
229	228	mptexd 7171	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})) ∈ V)
230	1, 227, 10, 229	fvmptd3 6962	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀‘𝐹) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})))
231	230	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → (𝑀‘𝐹) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})))
232		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 𝑖 ∈ (ℕ0 ↑m 1o))
233		fvexd 6845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}) ∈ V)
234	225, 231, 232, 233	fvmptd4 6963	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → ((𝑀‘𝐹)‘𝑖) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
235	221, 234	syldan 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑀‘𝐹)‘𝑖) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
236	235	adantlr 717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑀‘𝐹)‘𝑖) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
237		fveq1 6829	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝐺 → (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐺‘{⟨𝑋, (𝑛‘∅)⟩}))
238	237	mpteq2dv 5169	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐺 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})))
239	228	mptexd 7171	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})) ∈ V)
240	1, 238, 11, 239	fvmptd3 6962	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})))
241		fveq1 6829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑛‘∅) = (𝑚‘∅))
242	241	opeq2d 4814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ⟨𝑋, (𝑛‘∅)⟩ = ⟨𝑋, (𝑚‘∅)⟩)
243	242	sneqd 4570	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → {⟨𝑋, (𝑛‘∅)⟩} = {⟨𝑋, (𝑚‘∅)⟩})
244	243	fveq2d 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐺‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐺‘{⟨𝑋, (𝑚‘∅)⟩}))
245	244	cbvmptv 5179	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑚 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑚‘∅)⟩}))
246	240, 245	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑚 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑚‘∅)⟩})))
247	246	ad2antrr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑀‘𝐺) = (𝑚 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑚‘∅)⟩})))
248		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑚 = (𝑛 ∘f − 𝑖))
249	248	fveq1d 6832	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) = ((𝑛 ∘f − 𝑖)‘∅))
250	52	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ∅ ∈ 1o)
251	116	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛 Fn 1o)
252	251	ad2antrr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑛 Fn 1o)
253	100, 114	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 Fn 1o)
254	253	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑖 Fn 1o)
255	48	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 1o ∈ V)
256		eqidd 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ ∅ ∈ 1o) → (𝑛‘∅) = (𝑛‘∅))
257		eqidd 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ ∅ ∈ 1o) → (𝑖‘∅) = (𝑖‘∅))
258	252, 254, 255, 255, 119, 256, 257	ofval 7634	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ ∅ ∈ 1o) → ((𝑛 ∘f − 𝑖)‘∅) = ((𝑛‘∅) − (𝑖‘∅)))
259	250, 258	mpdan 689	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ((𝑛 ∘f − 𝑖)‘∅) = ((𝑛‘∅) − (𝑖‘∅)))
260	249, 259	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) = ((𝑛‘∅) − (𝑖‘∅)))
261	96	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑋 ∈ 𝑉)
262		fvexd 6845	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) ∈ V)
263		fvsng 7127	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑚‘∅) ∈ V) → ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (𝑚‘∅))
264	261, 262, 263	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (𝑚‘∅))
265	261, 152	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑋 ∈ {𝑋})
266	127	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
267	125	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑖‘∅)⟩} Fn {𝑋})
268	44	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {𝑋} ∈ V)
269	139	ad2antrr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ 𝑋 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
270	133	ad2antrr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ 𝑋 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
271	266, 267, 268, 268, 128, 269, 270	ofval 7634	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ 𝑋 ∈ {𝑋}) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋) = ((𝑛‘∅) − (𝑖‘∅)))
272	265, 271	mpdan 689	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋) = ((𝑛‘∅) − (𝑖‘∅)))
273	260, 264, 272	3eqtr4d 2781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋))
274		elsni 4575	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {(𝑛‘∅)} → 𝑥 = (𝑛‘∅))
275	274	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → 𝑥 = (𝑛‘∅))
276	275	oveq1d 7374	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → (𝑥 − 𝑦) = ((𝑛‘∅) − 𝑦))
277		fznn0sub2 13583	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0...(𝑛‘∅)) → ((𝑛‘∅) − 𝑦) ∈ (0...(𝑛‘∅)))
278	277	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → ((𝑛‘∅) − 𝑦) ∈ (0...(𝑛‘∅)))
279	276, 278	eqeltrd 2836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → (𝑥 − 𝑦) ∈ (0...(𝑛‘∅)))
280	279	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ (𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅)))) → (𝑥 − 𝑦) ∈ (0...(𝑛‘∅)))
281		fvex 6843	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛‘∅) ∈ V
282	150, 281	f1osn 6811	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {⟨∅, (𝑛‘∅)⟩}:{∅}–1-1-onto→{(𝑛‘∅)}
283		f1of 6770	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({⟨∅, (𝑛‘∅)⟩}:{∅}–1-1-onto→{(𝑛‘∅)} → {⟨∅, (𝑛‘∅)⟩}:{∅}⟶{(𝑛‘∅)})
284	282, 283	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩}:{∅}⟶{(𝑛‘∅)})
285		fvsng 7127	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∅ ∈ V ∧ (𝑛‘∅) ∈ ℕ0) → ({⟨∅, (𝑛‘∅)⟩}‘∅) = (𝑛‘∅))
286	150, 54, 285	sylancr 589	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ({⟨∅, (𝑛‘∅)⟩}‘∅) = (𝑛‘∅))
287	286	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛‘∅) = ({⟨∅, (𝑛‘∅)⟩}‘∅))
288	150	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ∅ ∈ V)
289	148	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {∅} = 1o)
290	53, 54	fsnd 6814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩}:{∅}⟶ℕ0)
291	289, 290	feq2dd 6644	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩}:1o⟶ℕ0)
292	291	ffnd 6659	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩} Fn 1o)
293	288, 147, 251, 292	fsneq 6979	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛 = {⟨∅, (𝑛‘∅)⟩} ↔ (𝑛‘∅) = ({⟨∅, (𝑛‘∅)⟩}‘∅)))
294	287, 293	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛 = {⟨∅, (𝑛‘∅)⟩})
295	147	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 1o = {∅})
296	294, 295	feq12d 6646	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛:1o⟶{(𝑛‘∅)} ↔ {⟨∅, (𝑛‘∅)⟩}:{∅}⟶{(𝑛‘∅)}))
297	284, 296	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛:1o⟶{(𝑛‘∅)})
298	297	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑛:1o⟶{(𝑛‘∅)})
299	147	fneq2i 6586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 Fn 1o ↔ 𝑖 Fn {∅})
300	253, 299	sylib 219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 Fn {∅})
301		0zd 12530	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 0 ∈ ℤ)
302	137	nn0zd 12543	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛‘∅) ∈ ℤ)
303	103	nn0zd 12543	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ∈ ℤ)
304	103	nn0ge0d 12495	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 0 ≤ (𝑖‘∅))
305	301, 302, 303, 304, 123	elfzd 13463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ∈ (0...(𝑛‘∅)))
306		fveq2 6830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑜 = ∅ → (𝑖‘𝑜) = (𝑖‘∅))
307	306	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑜 = ∅ → ((𝑖‘𝑜) ∈ (0...(𝑛‘∅)) ↔ (𝑖‘∅) ∈ (0...(𝑛‘∅))))
308	150, 307	ralsn 4616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑜 ∈ {∅} (𝑖‘𝑜) ∈ (0...(𝑛‘∅)) ↔ (𝑖‘∅) ∈ (0...(𝑛‘∅)))
309	305, 308	sylibr 235	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∀𝑜 ∈ {∅} (𝑖‘𝑜) ∈ (0...(𝑛‘∅)))
310		ffnfv 7063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖:{∅}⟶(0...(𝑛‘∅)) ↔ (𝑖 Fn {∅} ∧ ∀𝑜 ∈ {∅} (𝑖‘𝑜) ∈ (0...(𝑛‘∅))))
311	300, 309, 310	sylanbrc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖:{∅}⟶(0...(𝑛‘∅)))
312	147, 97	eqeltrrid 2841	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {∅} ∈ V)
313	147	ineq2i 4149	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1o ∩ 1o) = (1o ∩ {∅})
314	313, 119	eqtr3i 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1o ∩ {∅}) = 1o
315	280, 298, 311, 97, 312, 314	off 7641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛 ∘f − 𝑖):1o⟶(0...(𝑛‘∅)))
316		fz0ssnn0 13570	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...(𝑛‘∅)) ⊆ ℕ0
317	316	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (0...(𝑛‘∅)) ⊆ ℕ0)
318	315, 317	fssd 6675	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛 ∘f − 𝑖):1o⟶ℕ0)
319	318	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑛 ∘f − 𝑖):1o⟶ℕ0)
320	319, 250	ffvelcdmd 7029	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ((𝑛 ∘f − 𝑖)‘∅) ∈ ℕ0)
321	249, 320	eqeltrd 2836	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) ∈ ℕ0)
322	261, 321	fsnd 6814	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑚‘∅)⟩}:{𝑋}⟶ℕ0)
323	322	ffnd 6659	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑚‘∅)⟩} Fn {𝑋})
324	266, 267, 268, 268, 128	offn 7636	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}) Fn {𝑋})
325	261, 207, 323, 324	fsneq 6979	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑚‘∅)⟩} = ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}) ↔ ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋)))
326	273, 325	mpbird 258	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑚‘∅)⟩} = ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))
327	326	fveq2d 6834	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝐺‘{⟨𝑋, (𝑚‘∅)⟩}) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
328	94, 97, 318	elmapdd 8781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛 ∘f − 𝑖) ∈ (ℕ0 ↑m 1o))
329		fvexd 6845	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})) ∈ V)
330	247, 327, 328, 329	fvmptd 6946	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
331	236, 330	oveq12d 7377	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))) = ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))
332	331	mpteq2dva 5168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)))) = (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))))
333	332	oveq2d 7375	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))))
334	219, 333	eqtr4d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))))
335	19, 334	sylan9eqr 2793	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑚 = {⟨𝑋, (𝑛‘∅)⟩}) → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))))
336		ovexd 7394	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))) ∈ V)
337	13, 335, 62, 336	fvmptd 6946	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))))
338	337	mpteq2dva 5168	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)))))))
339		eqid 2736	. . . . 5 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
340		selvply1rhmlema.5	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝑅)
341		eqid 2736	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄)
342	340, 341	ply1bas 22183	. . . . 5 ⊢ (Base‘𝑄) = (Base‘(1o mPoly 𝑅))
343		selvply1rhmlema.4	. . . . . 6 ⊢ × = (.r‘𝑄)
344	340, 339, 343	ply1mulr 22213	. . . . 5 ⊢ × = (.r‘(1o mPoly 𝑅))
345		psr1baslem 22173	. . . . 5 ⊢ (ℕ0 ↑m 1o) = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
346	5, 4, 7, 343, 340, 1, 46, 27, 10	selvply1rhmlema 33705	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐹) ∈ (Base‘𝑄))
347	5, 4, 7, 343, 340, 1, 46, 27, 11	selvply1rhmlema 33705	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐺) ∈ (Base‘𝑄))
348	339, 342, 6, 344, 345, 346, 347	mplmul 21988	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐹) × (𝑀‘𝐺)) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)))))))
349	338, 348	eqtr4d 2774	. . 3 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩})) = ((𝑀‘𝐹) × (𝑀‘𝐺)))
350	3, 349	sylan9eqr 2793	. 2 ⊢ ((𝜑 ∧ 𝑓 = (𝐹 · 𝐺)) → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = ((𝑀‘𝐹) × (𝑀‘𝐺)))
351	44	a1i 11	. . . 4 ⊢ (𝜑 → {𝑋} ∈ V)
352	4, 351, 27	mplringd 22000	. . 3 ⊢ (𝜑 → 𝑃 ∈ Ring)
353	5, 7, 352, 10, 11	ringcld 20235	. 2 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
354		ovexd 7394	. 2 ⊢ (𝜑 → ((𝑀‘𝐹) × (𝑀‘𝐺)) ∈ V)
355	1, 350, 353, 354	fvmptd2 6947	1 ⊢ (𝜑 → (𝑀‘(𝐹 · 𝐺)) = ((𝑀‘𝐹) × (𝑀‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1543   ∈ wcel 2115  ∀wral 3050  {crab 3388  Vcvv 3428   ∩ cin 3885   ⊆ wss 3886  ∅c0 4264  {csn 4558  ⟨cop 4564   class class class wbr 5075   ↦ cmpt 5156   Fn wfn 6483  ⟶wf 6484  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7359   ∘_f cof 7621   ∘_r cofr 7622  1_oc1o 8391   ↑_m cmap 8766  Fincfn 8886   finSupp cfsupp 9267  0cc0 11032   ≤ cle 11174   − cmin 11371  ℕ₀cn0 12431  ...cfz 13455  Basecbs 17173  ._rcmulr 17215  0_gc0g 17396   Σ_g cgsu 17397  CMndccmn 19749  Ringcrg 20208   mPoly cmpl 21884  Poly₁cpl1 22165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7681  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3or 1089  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3061  df-rmo 3341  df-reu 3342  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3906  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-of 7623  df-ofr 7624  df-om 7810  df-1st 7934  df-2nd 7935  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-mulg 19038  df-subg 19093  df-ghm 19182  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-subrng 20521  df-subrg 20545  df-psr 21887  df-mpl 21889  df-opsr 21891  df-psr1 22168  df-ply1 22170

This theorem is referenced by:  selvply1rhm  33712