Theorem psrass1 22122

Description: Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)

Hypotheses

Ref	Expression
psrring.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrring.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrring.r	⊢ (𝜑 → 𝑅 ∈ Ring)
psrass.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
psrass.t	⊢ × = (.r‘𝑆)
psrass.b	⊢ 𝐵 = (Base‘𝑆)
psrass.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
psrass.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
psrass.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
psrass1	⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Distinct variable groups:   𝑓,𝐼   𝑅,𝑓   𝑓,𝑋   𝑓,𝑍   𝑓,𝑌

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐷(𝑓)   𝑆(𝑓)   × (𝑓)   𝑉(𝑓)

Proof of Theorem psrass1

Dummy variables 𝑥 𝑘 𝑧 𝑔 ℎ 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrring.s	. . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2763	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		psrass.d	. . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
4		psrass.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
5		psrass.t	. . . . 5 ⊢ × = (.r‘𝑆)
6		psrring.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		psrass.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		psrass.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 4, 5, 6, 7, 8	psrmulcl 22105	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10		psrass.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11	1, 4, 5, 6, 9, 10	psrmulcl 22105	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
12	1, 2, 3, 4, 11	psrelbas 22094	. . 3 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
13	12	ffnd 6706	. 2 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
14	1, 4, 5, 6, 8, 10	psrmulcl 22105	. . . . 5 ⊢ (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
15	1, 4, 5, 6, 7, 14	psrmulcl 22105	. . . 4 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
16	1, 2, 3, 4, 15	psrelbas 22094	. . 3 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
17	16	ffnd 6706	. 2 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18		eqid 2763	. . . . 5 ⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}
19		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
20	6	ringcmnd 20372	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
21	20	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ CMnd)
22		eqid 2763	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	6	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑅 ∈ Ring)
24	1, 2, 3, 4, 7	psrelbas 22094	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
25	24	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
26		breq1 5112	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑗 ∘r ≤ 𝑥))
27	26	elrab 3650	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥))
28	27	bilani 509	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥))
29	28	simpld 499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∈ 𝐷)
30	25, 29	ffvelcdmd 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
31	30	adantr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
32	1, 2, 3, 4, 8	psrelbas 22094	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
33	32	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
34		breq1 5112	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑛 → (ℎ ∘r ≤ (𝑥 ∘f − 𝑗) ↔ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
35	34	elrab 3650	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↔ (𝑛 ∈ 𝐷 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
36	35	bilani 509	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑛 ∈ 𝐷 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
37	36	simpld 499	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∈ 𝐷)
38	33, 37	ffvelcdmd 7080	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑌‘𝑛) ∈ (Base‘𝑅))
39	1, 2, 3, 4, 10	psrelbas 22094	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍:𝐷⟶(Base‘𝑅))
40	39	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
41		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷)
42	3	psrbagf 22077	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐷 → 𝑗:𝐼⟶ℕ0)
43	29, 42	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0)
44	28	simprd 500	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∘r ≤ 𝑥)
45	3	psrbagcon 22084	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥))
46	41, 43, 44, 45	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥))
47	46	simpld 499	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑗) ∈ 𝐷)
48	47	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑥 ∘f − 𝑗) ∈ 𝐷)
49	3	psrbagf 22077	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐷 → 𝑛:𝐼⟶ℕ0)
50	37, 49	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛:𝐼⟶ℕ0)
51	36	simprd 500	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗))
52	3	psrbagcon 22084	. . . . . . . . . . 11 ⊢ (((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ 𝑛:𝐼⟶ℕ0 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)) → (((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∈ 𝐷 ∧ ((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∘r ≤ (𝑥 ∘f − 𝑗)))
53	48, 50, 51, 52	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∈ 𝐷 ∧ ((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∘r ≤ (𝑥 ∘f − 𝑗)))
54	53	simpld 499	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∈ 𝐷)
55	40, 54	ffvelcdmd 7080	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)) ∈ (Base‘𝑅))
56	2, 22, 23, 38, 55	ringcld 20343	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))) ∈ (Base‘𝑅))
57	2, 22, 23, 31, 56	ringcld 20343	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ (Base‘𝑅))
58	57	anasss 471	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ (Base‘𝑅))
59		fveq2 6881	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑌‘𝑛) = (𝑌‘(𝑘 ∘f − 𝑗)))
60		oveq2 7418	. . . . . . . 8 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑥 ∘f − 𝑗) ∘f − 𝑛) = ((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))
61	60	fveq2d 6885	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)) = (𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))
62	59, 61	oveq12d 7428	. . . . . 6 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))) = ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))))
63	62	oveq2d 7426	. . . . 5 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))
64	3, 18, 19, 2, 21, 58, 63	psrass1lem 22092	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))))
65	7	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋 ∈ 𝐵)
66	8	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵)
67		breq1 5112	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑘 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑘 ∘r ≤ 𝑥))
68	67	elrab 3650	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥))
69	68	bilani 509	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥))
70	69	simpld 499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∈ 𝐷)
71	1, 4, 22, 5, 3, 65, 66, 70	psrmulval 22103	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))))))
72	71	oveq1d 7425	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))
73		eqid 2763	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
74	6	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring)
75	3	psrbaglefi 22085	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin)
76	70, 75	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin)
77	39	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
78		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷)
79	3	psrbagf 22077	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐷 → 𝑘:𝐼⟶ℕ0)
80	70, 79	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0)
81	69	simprd 500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∘r ≤ 𝑥)
82	3	psrbagcon 22084	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑘:𝐼⟶ℕ0 ∧ 𝑘 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥))
83	78, 80, 81, 82	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥))
84	83	simpld 499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑘) ∈ 𝐷)
85	77, 84	ffvelcdmd 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))
86	6	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑅 ∈ Ring)
87	24	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
88		breq1 5112	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑗 → (ℎ ∘r ≤ 𝑘 ↔ 𝑗 ∘r ≤ 𝑘))
89	88	elrab 3650	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘))
90	89	bilani 509	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘))
91	90	simpld 499	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∈ 𝐷)
92	87, 91	ffvelcdmd 7080	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
93	32	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
94	70	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷)
95	91, 42	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗:𝐼⟶ℕ0)
96	90	simprd 500	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∘r ≤ 𝑘)
97	3	psrbagcon 22084	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑘) → ((𝑘 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘f − 𝑗) ∘r ≤ 𝑘))
98	94, 95, 96, 97	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘f − 𝑗) ∘r ≤ 𝑘))
99	98	simpld 499	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ 𝐷)
100	93, 99	ffvelcdmd 7080	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅))
101	2, 22, 86, 92, 100	ringcld 20343	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))) ∈ (Base‘𝑅))
102		eqid 2763	. . . . . . . . 9 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))))
103		fvex 6894	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
104	103	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (0g‘𝑅) ∈ V)
105	102, 76, 101, 104	fsuppmptdm 9332	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))) finSupp (0g‘𝑅))
106	2, 73, 22, 74, 76, 85, 101, 105	gsummulc1 20402	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))
107	85	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))
108	2, 22	ringass 20339	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((𝑋‘𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅) ∧ (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))))
109	86, 92, 100, 107, 108	syl13anc 1399	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))))
110	3	psrbagf 22077	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐷 → 𝑥:𝐼⟶ℕ0)
111	110	ad3antlr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥:𝐼⟶ℕ0)
112	111	ffvelcdmda 7079	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈ ℕ0)
113	80	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
114	113	ffvelcdmda 7079	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
115	95	ffvelcdmda 7079	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈ ℕ0)
116		nn0cn 12518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ)
117		nn0cn 12518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
118		nn0cn 12518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ)
119		nnncan2 11499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥‘𝑧) ∈ ℂ ∧ (𝑘‘𝑧) ∈ ℂ ∧ (𝑗‘𝑧) ∈ ℂ) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
120	116, 117, 118, 119	syl3an 1178	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥‘𝑧) ∈ ℕ0 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑗‘𝑧) ∈ ℕ0) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
121	112, 114, 115, 120	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
122	121	mpteq2dva 5204	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧))))
123		psrring.i	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
124	123	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑉)
125		ovexd 7445	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V)
126		ovexd 7445	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧) − (𝑗‘𝑧)) ∈ V)
127	111	feqmptd 6949	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧)))
128	95	feqmptd 6949	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
129	124, 112, 115, 127, 128	offval2 7694	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑥 ∘f − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑗‘𝑧))))
130	113	feqmptd 6949	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
131	124, 114, 115, 130, 128	offval2 7694	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑗‘𝑧))))
132	124, 125, 126, 129, 131	offval2 7694	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)) = (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧)))))
133	124, 112, 114, 127, 130	offval2 7694	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑥 ∘f − 𝑘) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧))))
134	122, 132, 133	3eqtr4d 2808	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)) = (𝑥 ∘f − 𝑘))
135	134	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))) = (𝑍‘(𝑥 ∘f − 𝑘)))
136	135	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))) = ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))
137	136	oveq2d 7426	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))))
138	109, 137	eqtr4d 2801	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))
139	138	mpteq2dva 5204	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))))))
140	139	oveq2d 7426	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))
141	72, 106, 140	3eqtr2d 2804	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))
142	141	mpteq2dva 5204	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) = (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))))))))
143	142	oveq2d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))))
144	8	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵)
145	10	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍 ∈ 𝐵)
146	1, 4, 22, 5, 3, 144, 145, 47	psrmulval 22103	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))))
147	146	oveq2d 7426	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))
148	6	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring)
149	3	psrbaglefi 22085	. . . . . . . . 9 ⊢ ((𝑥 ∘f − 𝑗) ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin)
150	47, 149	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin)
151		ovex 7443	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m 𝐼) ∈ V
152	3, 151	rab2ex 5312	. . . . . . . . . . . 12 ⊢ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ V
153	152	mptex 7221	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V
154		funmpt 6574	. . . . . . . . . . 11 ⊢ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))
155	153, 154, 103	3pm3.2i 1358	. . . . . . . . . 10 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∧ (0g‘𝑅) ∈ V)
156	155	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∧ (0g‘𝑅) ∈ V))
157		suppssdm 8169	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ dom (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))
158		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) = (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))
159	158	dmmptss 6242	. . . . . . . . . . 11 ⊢ dom (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}
160	157, 159	sstri 3946	. . . . . . . . . 10 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}
161	160	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})
162		suppssfifsupp 9336	. . . . . . . . 9 ⊢ ((((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∧ (0g‘𝑅) ∈ V) ∧ ({ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin ∧ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) finSupp (0g‘𝑅))
163	156, 150, 161, 162	syl12anc 849	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) finSupp (0g‘𝑅))
164	2, 73, 22, 148, 150, 30, 56, 163	gsummulc2 20403	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))
165	147, 164	eqtr4d 2801	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))
166	165	mpteq2dva 5204	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)))) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))))))
167	166	oveq2d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))))
168	64, 143, 167	3eqtr4d 2808	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))))))
169	9	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
170	10	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ∈ 𝐵)
171	1, 4, 22, 5, 3, 169, 170, 19	psrmulval 22103	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))))
172	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑋 ∈ 𝐵)
173	14	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
174	1, 4, 22, 5, 3, 172, 173, 19	psrmulval 22103	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))))))
175	168, 171, 174	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
176	13, 17, 175	eqfnfvd 7028	1 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ⊆ wss 3905   class class class wbr 5109   ↦ cmpt 5192  ^◡ccnv 5660  dom cdm 5661   “ cima 5664  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7672   ∘_r cofr 7673   supp csupp 8152   ↑_m cmap 8820  Fincfn 8939   finSupp cfsupp 9317  ℂcc 11102   ≤ cle 11248   − cmin 11445  ℕcn 12237  ℕ₀cn0 12508  Basecbs 17273  ._rcmulr 17315  0_gc0g 17496   Σ_g cgsu 17497  CMndccmn 19854  Ringcrg 20319   mPwSer cmps 22063

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-oi 9468  df-card 9930  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-2 12307  df-3 12308  df-4 12309  df-5 12310  df-6 12311  df-7 12312  df-8 12313  df-9 12314  df-n0 12509  df-z 12596  df-uz 12867  df-fz 13540  df-fzo 13688  df-seq 14043  df-hash 14372  df-struct 17211  df-sets 17228  df-slot 17246  df-ndx 17258  df-base 17274  df-ress 17295  df-plusg 17327  df-mulr 17328  df-sca 17330  df-vsca 17331  df-tset 17333  df-0g 17498  df-gsum 17499  df-mre 17642  df-mrc 17643  df-acs 17645  df-mgm 18702  df-sgrp 18781  df-mnd 18797  df-mhm 18845  df-submnd 18846  df-grp 19007  df-minusg 19008  df-mulg 19138  df-ghm 19288  df-cntz 19391  df-cmn 19856  df-abl 19857  df-mgp 20221  df-ur 20268  df-ring 20321  df-psr 22068

This theorem is used by:  psrring  22128