Theorem mplvrpmga 33999

Description: The action of permuting variables in a multivariate polynomial is a group action. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
mplvrpmga.1	⊢ 𝑆 = (SymGrp‘𝐼)
mplvrpmga.2	⊢ 𝑃 = (Base‘𝑆)
mplvrpmga.3	⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))
mplvrpmga.4	⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
mplvrpmga.5	⊢ (𝜑 → 𝐼 ∈ 𝑉)

Assertion

Ref	Expression
mplvrpmga	⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐼,𝑑,𝑓,ℎ,𝑥   𝑀,𝑑,𝑓,𝑥   𝑃,𝑑,𝑓,𝑥   𝑥,𝑅   𝜑,𝑑,𝑓,𝑥

Allowed substitution hints:   𝜑(ℎ)   𝐴(ℎ)   𝑃(ℎ)   𝑅(𝑓, ℎ, 𝑑)   𝑆(𝑥, 𝑓, ℎ, 𝑑)   𝑀(ℎ)   𝑉(𝑥, 𝑓, ℎ, 𝑑)

Proof of Theorem mplvrpmga

Dummy variables 𝑔 𝑝 𝑞 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplvrpmga.5	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2		mplvrpmga.1	. . . 4 ⊢ 𝑆 = (SymGrp‘𝐼)
3	2	symggrp 19514	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑆 ∈ Grp)
4	1, 3	syl 18	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
5		mplvrpmga.3	. . . 4 ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))
6	5	fvexi 6899	. . 3 ⊢ 𝑀 ∈ V
7	6	a1i 11	. 2 ⊢ (𝜑 → 𝑀 ∈ V)
8		fvexd 6900	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (Base‘𝑅) ∈ V)
9		ovex 7452	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
10	9	rabex 5311	. . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V)
12		eqid 2765	. . . . . . . . 9 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
13		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2765	. . . . . . . . . 10 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
15	14	psrbasfsupp 33965	. . . . . . . . 9 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
16		xp2nd 8025	. . . . . . . . . 10 ⊢ (𝑐 ∈ (𝑃 × 𝑀) → (2nd ‘𝑐) ∈ 𝑀)
17	16	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (2nd ‘𝑐) ∈ 𝑀)
18	12, 13, 5, 15, 17	mplelf 22197	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (2nd ‘𝑐):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
19		breq1 5114	. . . . . . . . 9 ⊢ (ℎ = (𝑥 ∘ (1st ‘𝑐)) → (ℎ finSupp 0 ↔ (𝑥 ∘ (1st ‘𝑐)) finSupp 0))
20		nn0ex 12525	. . . . . . . . . . 11 ⊢ ℕ0 ∈ V
21	20	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
22	1	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
23		ssrab2 4035	. . . . . . . . . . . . . 14 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼)
24	23	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
25	24	sselda 3938	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ (ℕ0 ↑m 𝐼))
26	22, 21, 25	elmaprd 33096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥:𝐼⟶ℕ0)
27		xp1st 8024	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝑃 × 𝑀) → (1st ‘𝑐) ∈ 𝑃)
28	27	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1st ‘𝑐) ∈ 𝑃)
29		mplvrpmga.2	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Base‘𝑆)
30	2, 29	symgbasf 19490	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑐) ∈ 𝑃 → (1st ‘𝑐):𝐼⟶𝐼)
31	28, 30	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1st ‘𝑐):𝐼⟶𝐼)
32	26, 31	fcod 6735	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)):𝐼⟶ℕ0)
33	21, 22, 32	elmapdd 8844	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)) ∈ (ℕ0 ↑m 𝐼))
34		breq1 5114	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑥 → (ℎ finSupp 0 ↔ 𝑥 finSupp 0))
35	34	elrab 3652	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝑥 ∈ (ℕ0 ↑m 𝐼) ∧ 𝑥 finSupp 0))
36	35	bilani 510	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∈ (ℕ0 ↑m 𝐼) ∧ 𝑥 finSupp 0))
37	36	simprd 501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 finSupp 0)
38	2, 29	symgbasf1o 19489	. . . . . . . . . . 11 ⊢ ((1st ‘𝑐) ∈ 𝑃 → (1st ‘𝑐):𝐼–1-1-onto→𝐼)
39		f1of1 6823	. . . . . . . . . . 11 ⊢ ((1st ‘𝑐):𝐼–1-1-onto→𝐼 → (1st ‘𝑐):𝐼–1-1→𝐼)
40	28, 38, 39	3syl 19	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1st ‘𝑐):𝐼–1-1→𝐼)
41		0nn0 12534	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
42	41	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
43		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
44	37, 40, 42, 43	fsuppco 9369	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)) finSupp 0)
45	19, 33, 44	elrabd 3654	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
46	18, 45	ffvelcdmd 7084	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))) ∈ (Base‘𝑅))
47	46	fmpttd 7114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
48	8, 11, 47	elmapdd 8844	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
49		eqid 2765	. . . . . . 7 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
50		eqid 2765	. . . . . . 7 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
51	49, 13, 15, 50, 1	psrbas 22134	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
52	51	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
53	48, 52	eleqtrrd 2868	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ (Base‘(𝐼 mPwSer 𝑅)))
54		coeq1 5845	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∘ (1st ‘𝑐)) = (𝑦 ∘ (1st ‘𝑐)))
55	54	fveq2d 6889	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))) = ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐))))
56	55	cbvmptv 5217	. . . . 5 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐))))
57		fveq1 6884	. . . . . . . 8 ⊢ (𝑔 = (2nd ‘𝑐) → (𝑔‘(𝑦 ∘ 𝑞)) = ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞)))
58	57	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑔 = (2nd ‘𝑐) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))))
59	58	breq1d 5121	. . . . . 6 ⊢ (𝑔 = (2nd ‘𝑐) → ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅) ↔ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅)))
60		coeq2 5846	. . . . . . . . 9 ⊢ (𝑞 = (1st ‘𝑐) → (𝑦 ∘ 𝑞) = (𝑦 ∘ (1st ‘𝑐)))
61	60	fveq2d 6889	. . . . . . . 8 ⊢ (𝑞 = (1st ‘𝑐) → ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞)) = ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐))))
62	61	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑞 = (1st ‘𝑐) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐)))))
63	62	breq1d 5121	. . . . . 6 ⊢ (𝑞 = (1st ‘𝑐) → ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅) ↔ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅)))
64		mplvrpmga.4	. . . . . . . . . . . . 13 ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
65	64	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))
66		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → 𝑓 = 𝑔)
67		coeq2 5846	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑞 → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑞))
68	67	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑞))
69	66, 68	fveq12d 6892	. . . . . . . . . . . . . 14 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘(𝑥 ∘ 𝑞)))
70	69	mpteq2dv 5207	. . . . . . . . . . . . 13 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))))
71	70	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) ∧ (𝑑 = 𝑞 ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))))
72		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝑞 ∈ 𝑃)
73		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝑔 ∈ 𝑀)
74	10	mptex 7228	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))) ∈ V
75	74	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))) ∈ V)
76	65, 71, 72, 73, 75	ovmpod 7571	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))))
77		coeq1 5845	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∘ 𝑞) = (𝑦 ∘ 𝑞))
78	77	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑔‘(𝑥 ∘ 𝑞)) = (𝑔‘(𝑦 ∘ 𝑞)))
79	78	cbvmptv 5217	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))
80	76, 79	eqtrdi 2816	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
81	1	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝐼 ∈ 𝑉)
82		eqid 2765	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
83	2, 29, 5, 64, 81, 82, 73, 72	mplvrpmfgalem 33998	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) finSupp (0g‘𝑅))
84	80, 83	eqbrtrrd 5137	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
85	84	anasss 472	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑀 ∧ 𝑞 ∈ 𝑃)) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
86	85	ralrimivva 3210	. . . . . . 7 ⊢ (𝜑 → ∀𝑔 ∈ 𝑀 ∀𝑞 ∈ 𝑃 (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
87	86	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → ∀𝑔 ∈ 𝑀 ∀𝑞 ∈ 𝑃 (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
88	16	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (2nd ‘𝑐) ∈ 𝑀)
89	27	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (1st ‘𝑐) ∈ 𝑃)
90	59, 63, 87, 88, 89	rspc2dv 3598	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅))
91	56, 90	eqbrtrid 5148	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅))
92	12, 49, 50, 82, 5	mplelbas 22190	. . . 4 ⊢ ((𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ 𝑀 ↔ ((𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅)))
93	53, 91, 92	sylanbrc 595	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ 𝑀)
94		vex 3461	. . . . . . . 8 ⊢ 𝑑 ∈ V
95		vex 3461	. . . . . . . 8 ⊢ 𝑓 ∈ V
96	94, 95	op2ndd 8003	. . . . . . 7 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (2nd ‘𝑐) = 𝑓)
97	94, 95	op1std 8002	. . . . . . . 8 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (1st ‘𝑐) = 𝑑)
98	97	coeq2d 5850	. . . . . . 7 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (𝑥 ∘ (1st ‘𝑐)) = (𝑥 ∘ 𝑑))
99	96, 98	fveq12d 6892	. . . . . 6 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))) = (𝑓‘(𝑥 ∘ 𝑑)))
100	99	mpteq2dv 5207	. . . . 5 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
101	100	mpompt 7533	. . . 4 ⊢ (𝑐 ∈ (𝑃 × 𝑀) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))))) = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
102	64, 101	eqtr4i 2791	. . 3 ⊢ 𝐴 = (𝑐 ∈ (𝑃 × 𝑀) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))))
103	93, 102	fmptd 7113	. 2 ⊢ (𝜑 → 𝐴:(𝑃 × 𝑀)⟶𝑀)
104	2	symgid 19515	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → ( I ↾ 𝐼) = (0g‘𝑆))
105	1, 104	syl 18	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐼) = (0g‘𝑆))
106	105	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ( I ↾ 𝐼) = (0g‘𝑆))
107	106	oveq1d 7434	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (( I ↾ 𝐼)𝐴𝑔) = ((0g‘𝑆)𝐴𝑔))
108	64	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))
109	1	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
110	20	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
111	23	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
112	111	sselda 3938	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ (ℕ0 ↑m 𝐼))
113	109, 110, 112	elmaprd 33096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥:𝐼⟶ℕ0)
114		fcoi1 6756	. . . . . . . . . . 11 ⊢ (𝑥:𝐼⟶ℕ0 → (𝑥 ∘ ( I ↾ 𝐼)) = 𝑥)
115	113, 114	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ ( I ↾ 𝐼)) = 𝑥)
116	115	fveq2d 6889	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘(𝑥 ∘ ( I ↾ 𝐼))) = (𝑔‘𝑥))
117	116	mpteq2dva 5206	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
118	117	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
119		simpr 490	. . . . . . . . . 10 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → 𝑓 = 𝑔)
120		coeq2 5846	. . . . . . . . . . 11 ⊢ (𝑑 = ( I ↾ 𝐼) → (𝑥 ∘ 𝑑) = (𝑥 ∘ ( I ↾ 𝐼)))
121	120	adantr 486	. . . . . . . . . 10 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → (𝑥 ∘ 𝑑) = (𝑥 ∘ ( I ↾ 𝐼)))
122	119, 121	fveq12d 6892	. . . . . . . . 9 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘(𝑥 ∘ ( I ↾ 𝐼))))
123	122	mpteq2dv 5207	. . . . . . . 8 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))))
124	123	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))))
125	12, 49, 50, 82, 5	mplelbas 22190	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑀 ↔ (𝑔 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑔 finSupp (0g‘𝑅)))
126	125	simplbi 502	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑀 → 𝑔 ∈ (Base‘(𝐼 mPwSer 𝑅)))
127	49, 13, 15, 50, 126	psrelbas 22135	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑀 → 𝑔:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
128	127	ad3antlr 744	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑑 = ( I ↾ 𝐼)) ∧ 𝑓 = 𝑔) → 𝑔:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
129	128	feqmptd 6953	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑑 = ( I ↾ 𝐼)) ∧ 𝑓 = 𝑔) → 𝑔 = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
130	129	anasss 472	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → 𝑔 = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
131	118, 124, 130	3eqtr4d 2810	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = 𝑔)
132		eqid 2765	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
133	29, 132	grpidcl 19076	. . . . . . . . 9 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝑃)
134	1, 3, 133	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝑃)
135	105, 134	eqeltrd 2865	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐼) ∈ 𝑃)
136	135	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ( I ↾ 𝐼) ∈ 𝑃)
137		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ 𝑀)
138	108, 131, 136, 137, 137	ovmpod 7571	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (( I ↾ 𝐼)𝐴𝑔) = 𝑔)
139	107, 138	eqtr3d 2802	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ((0g‘𝑆)𝐴𝑔) = 𝑔)
140		eqid 2765	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
141	2, 29, 140	symgov 19498	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝑞 ∈ 𝑃) → (𝑝(+g‘𝑆)𝑞) = (𝑝 ∘ 𝑞))
142	141	adantll 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝(+g‘𝑆)𝑞) = (𝑝 ∘ 𝑞))
143	142	oveq1d 7434	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = ((𝑝 ∘ 𝑞)𝐴𝑔))
144		coass 6269	. . . . . . . . . . 11 ⊢ ((𝑥 ∘ 𝑝) ∘ 𝑞) = (𝑥 ∘ (𝑝 ∘ 𝑞))
145	144	a1i 11	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((𝑥 ∘ 𝑝) ∘ 𝑞) = (𝑥 ∘ (𝑝 ∘ 𝑞)))
146	145	fveq2d 6889	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)) = (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞))))
147	146	mpteq2dva 5206	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
148	80	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
149	148	oveq2d 7435	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝𝐴(𝑞𝐴𝑔)) = (𝑝𝐴(𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))))
150	64	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))
151		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑑 = 𝑝)
152	151	coeq2d 5850	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑝))
153	152	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑓‘(𝑥 ∘ 𝑝)))
154		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
155		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑦 = (𝑥 ∘ 𝑝)) → 𝑦 = (𝑥 ∘ 𝑝))
156	155	coeq1d 5849	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑦 = (𝑥 ∘ 𝑝)) → (𝑦 ∘ 𝑞) = ((𝑥 ∘ 𝑝) ∘ 𝑞))
157	156	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑦 = (𝑥 ∘ 𝑝)) → (𝑔‘(𝑦 ∘ 𝑞)) = (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)))
158		breq1 5114	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑥 ∘ 𝑝) → (ℎ finSupp 0 ↔ (𝑥 ∘ 𝑝) finSupp 0))
159	20	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
160	1	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝐼 ∈ 𝑉)
161	160	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
162	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
163	162	sselda 3938	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ (ℕ0 ↑m 𝐼))
164	161, 159, 163	elmaprd 33096	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥:𝐼⟶ℕ0)
165	2, 29	symgbasf 19490	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑃 → 𝑝:𝐼⟶𝐼)
166	165	ad5antlr 748	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑝:𝐼⟶𝐼)
167	164, 166	fcod 6735	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝):𝐼⟶ℕ0)
168	159, 161, 167	elmapdd 8844	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝) ∈ (ℕ0 ↑m 𝐼))
169	35	simprbi 503	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} → 𝑥 finSupp 0)
170	169	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 finSupp 0)
171	2, 29	symgbasf1o 19489	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑃 → 𝑝:𝐼–1-1-onto→𝐼)
172		f1of1 6823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝:𝐼–1-1-onto→𝐼 → 𝑝:𝐼–1-1→𝐼)
173	171, 172	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ 𝑃 → 𝑝:𝐼–1-1→𝐼)
174	173	ad5antlr 748	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑝:𝐼–1-1→𝐼)
175	41	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
176		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
177	170, 174, 175, 176	fsuppco 9369	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝) finSupp 0)
178	158, 168, 177	elrabd 3654	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
179		fvexd 6900	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)) ∈ V)
180		nfv 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝)
181		nfmpt1 5212	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦(𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))
182	181	nfeq2 2944	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))
183	180, 182	nfan 1932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
184		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
185	183, 184	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
186		nfcv 2927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝑥 ∘ 𝑝)
187		nfcv 2927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))
188	154, 157, 178, 179, 185, 186, 187	fvmptdf 7000	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑓‘(𝑥 ∘ 𝑝)) = (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)))
189	153, 188	eqtrd 2800	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)))
190	189	mpteq2dva 5206	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
191	190	anasss 472	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ (𝑑 = 𝑝 ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
192		simplr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑝 ∈ 𝑃)
193		fvexd 6900	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (Base‘𝑅) ∈ V)
194	10	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V)
195	137	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑔 ∈ 𝑀)
196	12, 13, 5, 15, 195	mplelf 22197	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑔:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
197		breq1 5114	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑦 ∘ 𝑞) → (ℎ finSupp 0 ↔ (𝑦 ∘ 𝑞) finSupp 0))
198	20	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
199	160	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
200	23	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
201	200	sselda 3938	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦 ∈ (ℕ0 ↑m 𝐼))
202	199, 198, 201	elmaprd 33096	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦:𝐼⟶ℕ0)
203	2, 29	symgbasf 19490	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ 𝑃 → 𝑞:𝐼⟶𝐼)
204	203	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑞:𝐼⟶𝐼)
205	202, 204	fcod 6735	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞):𝐼⟶ℕ0)
206	198, 199, 205	elmapdd 8844	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞) ∈ (ℕ0 ↑m 𝐼))
207		breq1 5114	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝑦 → (ℎ finSupp 0 ↔ 𝑦 finSupp 0))
208	207	elrab 3652	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝑦 ∈ (ℕ0 ↑m 𝐼) ∧ 𝑦 finSupp 0))
209	208	simprbi 503	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} → 𝑦 finSupp 0)
210	209	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦 finSupp 0)
211	2, 29	symgbasf1o 19489	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ 𝑃 → 𝑞:𝐼–1-1-onto→𝐼)
212	211	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑞:𝐼–1-1-onto→𝐼)
213		f1of1 6823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞:𝐼–1-1-onto→𝐼 → 𝑞:𝐼–1-1→𝐼)
214	212, 213	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑞:𝐼–1-1→𝐼)
215	41	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
216		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
217	210, 214, 215, 216	fsuppco 9369	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞) finSupp 0)
218	197, 206, 217	elrabd 3654	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
219	196, 218	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘(𝑦 ∘ 𝑞)) ∈ (Base‘𝑅))
220	219	fmpttd 7114	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
221	193, 194, 220	elmapdd 8844	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
222	51	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
223	221, 222	eleqtrrd 2868	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ (Base‘(𝐼 mPwSer 𝑅)))
224	84	adantlr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
225	12, 49, 50, 82, 5	mplelbas 22190	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ 𝑀 ↔ ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅)))
226	223, 224, 225	sylanbrc 595	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ 𝑀)
227	194	mptexd 7229	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))) ∈ V)
228	150, 191, 192, 226, 227	ovmpod 7571	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝𝐴(𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
229	149, 228	eqtrd 2800	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝𝐴(𝑞𝐴𝑔)) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
230		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → 𝑓 = 𝑔)
231		coeq2 5846	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑝 ∘ 𝑞) → (𝑥 ∘ 𝑑) = (𝑥 ∘ (𝑝 ∘ 𝑞)))
232	231	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → (𝑥 ∘ 𝑑) = (𝑥 ∘ (𝑝 ∘ 𝑞)))
233	230, 232	fveq12d 6892	. . . . . . . . . . 11 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞))))
234	233	mpteq2dv 5207	. . . . . . . . . 10 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
235	234	adantl 487	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ (𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
236	160, 3	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑆 ∈ Grp)
237		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑞 ∈ 𝑃)
238	29, 140, 236, 192, 237	grpcld 19058	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝(+g‘𝑆)𝑞) ∈ 𝑃)
239	142, 238	eqeltrrd 2866	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝 ∘ 𝑞) ∈ 𝑃)
240		simpllr 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑔 ∈ 𝑀)
241	194	mptexd 7229	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))) ∈ V)
242	150, 235, 239, 240, 241	ovmpod 7571	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝 ∘ 𝑞)𝐴𝑔) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
243	147, 229, 242	3eqtr4rd 2811	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝 ∘ 𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
244	143, 243	eqtrd 2800	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
245	244	anasss 472	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑝 ∈ 𝑃 ∧ 𝑞 ∈ 𝑃)) → ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
246	245	ralrimivva 3210	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
247	139, 246	jca 521	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (((0g‘𝑆)𝐴𝑔) = 𝑔 ∧ ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔))))
248	247	ralrimiva 3159	. 2 ⊢ (𝜑 → ∀𝑔 ∈ 𝑀 (((0g‘𝑆)𝐴𝑔) = 𝑔 ∧ ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔))))
249	29, 140, 132	isga 19405	. 2 ⊢ (𝐴 ∈ (𝑆 GrpAct 𝑀) ↔ ((𝑆 ∈ Grp ∧ 𝑀 ∈ V) ∧ (𝐴:(𝑃 × 𝑀)⟶𝑀 ∧ ∀𝑔 ∈ 𝑀 (((0g‘𝑆)𝐴𝑔) = 𝑔 ∧ ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔))))))
250	4, 7, 103, 248, 249	syl22anbrc 32877	1 ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  Vcvv 3457   ⊆ wss 3906  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   I cid 5557   × cxp 5661   ↾ cres 5665   ∘ ccom 5667  ⟶wf 6536  –1-1→wf1 6537  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991   ↑_m cmap 8830   finSupp cfsupp 9328  0cc0 11115  ℕ₀cn0 12519  Basecbs 17291  +_gcplusg 17332  0_gc0g 17514  Grpcgrp 19044   GrpAct cga 19403  SymGrpcsymg 19483   mPwSer cmps 22104   mPoly cmpl 22106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-uz 12879  df-fz 13552  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-sca 17348  df-vsca 17349  df-tset 17351  df-0g 17516  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-submnd 18879  df-efmnd 18965  df-grp 19047  df-ga 19404  df-symg 19484  df-psr 22109  df-mpl 22111

This theorem is used by:  mplvrpmmhm  34000  mplvrpmrhm  34001  splysubrg  34014  issply  34015