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Theorem psrass1 21374
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 (𝜑𝐼𝑉)
psrring.r (𝜑𝑅 ∈ Ring)
psrass.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrass.t × = (.r𝑆)
psrass.b 𝐵 = (Base‘𝑆)
psrass.x (𝜑𝑋𝐵)
psrass.y (𝜑𝑌𝐵)
psrass.z (𝜑𝑍𝐵)
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 𝑗 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrring.s . . . 4 𝑆 = (𝐼 mPwSer 𝑅)
2 eqid 2736 . . . 4 (Base‘𝑅) = (Base‘𝑅)
3 psrass.d . . . 4 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
4 psrass.b . . . 4 𝐵 = (Base‘𝑆)
5 psrass.t . . . . 5 × = (.r𝑆)
6 psrring.r . . . . 5 (𝜑𝑅 ∈ Ring)
7 psrass.x . . . . . 6 (𝜑𝑋𝐵)
8 psrass.y . . . . . 6 (𝜑𝑌𝐵)
91, 4, 5, 6, 7, 8psrmulcl 21356 . . . . 5 (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10 psrass.z . . . . 5 (𝜑𝑍𝐵)
111, 4, 5, 6, 9, 10psrmulcl 21356 . . . 4 (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
121, 2, 3, 4, 11psrelbas 21347 . . 3 (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
1312ffnd 6669 . 2 (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
141, 4, 5, 6, 8, 10psrmulcl 21356 . . . . 5 (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
151, 4, 5, 6, 7, 14psrmulcl 21356 . . . 4 (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
161, 2, 3, 4, 15psrelbas 21347 . . 3 (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
1716ffnd 6669 . 2 (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18 eqid 2736 . . . . 5 {𝑔𝐷𝑔r𝑥} = {𝑔𝐷𝑔r𝑥}
19 simpr 485 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝐷)
20 ringcmn 20003 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
216, 20syl 17 . . . . . 6 (𝜑𝑅 ∈ CMnd)
2221adantr 481 . . . . 5 ((𝜑𝑥𝐷) → 𝑅 ∈ CMnd)
236ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
2423adantr 481 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑅 ∈ Ring)
251, 2, 3, 4, 7psrelbas 21347 . . . . . . . . . 10 (𝜑𝑋:𝐷⟶(Base‘𝑅))
2625ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
27 simpr 485 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗 ∈ {𝑔𝐷𝑔r𝑥})
28 breq1 5108 . . . . . . . . . . . 12 (𝑔 = 𝑗 → (𝑔r𝑥𝑗r𝑥))
2928elrab 3645 . . . . . . . . . . 11 (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑗𝐷𝑗r𝑥))
3027, 29sylib 217 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗𝐷𝑗r𝑥))
3130simpld 495 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗𝐷)
3226, 31ffvelcdmd 7036 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑋𝑗) ∈ (Base‘𝑅))
3332adantr 481 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑋𝑗) ∈ (Base‘𝑅))
341, 2, 3, 4, 8psrelbas 21347 . . . . . . . . . 10 (𝜑𝑌:𝐷⟶(Base‘𝑅))
3534ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
36 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})
37 breq1 5108 . . . . . . . . . . . 12 ( = 𝑛 → (r ≤ (𝑥f𝑗) ↔ 𝑛r ≤ (𝑥f𝑗)))
3837elrab 3645 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↔ (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
3936, 38sylib 217 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
4039simpld 495 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛𝐷)
4135, 40ffvelcdmd 7036 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑌𝑛) ∈ (Base‘𝑅))
421, 2, 3, 4, 10psrelbas 21347 . . . . . . . . . 10 (𝜑𝑍:𝐷⟶(Base‘𝑅))
4342ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
44 simplr 767 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
453psrbagf 21320 . . . . . . . . . . . . . . 15 (𝑗𝐷𝑗:𝐼⟶ℕ0)
4631, 45syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗:𝐼⟶ℕ0)
4730simprd 496 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗r𝑥)
483psrbagcon 21332 . . . . . . . . . . . . . 14 ((𝑥𝐷𝑗:𝐼⟶ℕ0𝑗r𝑥) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
4944, 46, 47, 48syl3anc 1371 . . . . . . . . . . . . 13 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
5049simpld 495 . . . . . . . . . . . 12 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑗) ∈ 𝐷)
5150adantr 481 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑥f𝑗) ∈ 𝐷)
523psrbagf 21320 . . . . . . . . . . . 12 (𝑛𝐷𝑛:𝐼⟶ℕ0)
5340, 52syl 17 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛:𝐼⟶ℕ0)
5439simprd 496 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛r ≤ (𝑥f𝑗))
553psrbagcon 21332 . . . . . . . . . . 11 (((𝑥f𝑗) ∈ 𝐷𝑛:𝐼⟶ℕ0𝑛r ≤ (𝑥f𝑗)) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
5651, 53, 54, 55syl3anc 1371 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
5756simpld 495 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑥f𝑗) ∘f𝑛) ∈ 𝐷)
5843, 57ffvelcdmd 7036 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅))
59 eqid 2736 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
602, 59ringcl 19981 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑌𝑛) ∈ (Base‘𝑅) ∧ (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅)) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
6124, 41, 58, 60syl3anc 1371 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
622, 59ringcl 19981 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑋𝑗) ∈ (Base‘𝑅) ∧ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅)) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
6324, 33, 61, 62syl3anc 1371 . . . . . 6 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
6463anasss 467 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
65 fveq2 6842 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑌𝑛) = (𝑌‘(𝑘f𝑗)))
66 oveq2 7365 . . . . . . . 8 (𝑛 = (𝑘f𝑗) → ((𝑥f𝑗) ∘f𝑛) = ((𝑥f𝑗) ∘f − (𝑘f𝑗)))
6766fveq2d 6846 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) = (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))
6865, 67oveq12d 7375 . . . . . 6 (𝑛 = (𝑘f𝑗) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))
6968oveq2d 7373 . . . . 5 (𝑛 = (𝑘f𝑗) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
703, 18, 19, 2, 22, 64, 69psrass1lem 21345 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))))
717ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋𝐵)
728ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
73 simpr 485 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘 ∈ {𝑔𝐷𝑔r𝑥})
74 breq1 5108 . . . . . . . . . . . 12 (𝑔 = 𝑘 → (𝑔r𝑥𝑘r𝑥))
7574elrab 3645 . . . . . . . . . . 11 (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑘𝐷𝑘r𝑥))
7673, 75sylib 217 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑘𝐷𝑘r𝑥))
7776simpld 495 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘𝐷)
781, 4, 59, 5, 3, 71, 72, 77psrmulval 21354 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))))
7978oveq1d 7372 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))))(.r𝑅)(𝑍‘(𝑥f𝑘))))
80 eqid 2736 . . . . . . . 8 (0g𝑅) = (0g𝑅)
81 eqid 2736 . . . . . . . 8 (+g𝑅) = (+g𝑅)
826ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
833psrbaglefi 21334 . . . . . . . . 9 (𝑘𝐷 → {𝐷r𝑘} ∈ Fin)
8477, 83syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r𝑘} ∈ Fin)
8542ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
86 simplr 767 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
873psrbagf 21320 . . . . . . . . . . . 12 (𝑘𝐷𝑘:𝐼⟶ℕ0)
8877, 87syl 17 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘:𝐼⟶ℕ0)
8976simprd 496 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘r𝑥)
903psrbagcon 21332 . . . . . . . . . . 11 ((𝑥𝐷𝑘:𝐼⟶ℕ0𝑘r𝑥) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
9186, 88, 89, 90syl3anc 1371 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
9291simpld 495 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑘) ∈ 𝐷)
9385, 92ffvelcdmd 7036 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
9482adantr 481 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑅 ∈ Ring)
9525ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
96 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 ∈ {𝐷r𝑘})
97 breq1 5108 . . . . . . . . . . . . 13 ( = 𝑗 → (r𝑘𝑗r𝑘))
9897elrab 3645 . . . . . . . . . . . 12 (𝑗 ∈ {𝐷r𝑘} ↔ (𝑗𝐷𝑗r𝑘))
9996, 98sylib 217 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑗𝐷𝑗r𝑘))
10099simpld 495 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗𝐷)
10195, 100ffvelcdmd 7036 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑋𝑗) ∈ (Base‘𝑅))
10234ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
10377adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘𝐷)
104100, 45syl 17 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗:𝐼⟶ℕ0)
10599simprd 496 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗r𝑘)
1063psrbagcon 21332 . . . . . . . . . . . 12 ((𝑘𝐷𝑗:𝐼⟶ℕ0𝑗r𝑘) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
107103, 104, 105, 106syl3anc 1371 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
108107simpld 495 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
109102, 108ffvelcdmd 7036 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅))
1102, 59ringcl 19981 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑋𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅)) → ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))) ∈ (Base‘𝑅))
11194, 101, 109, 110syl3anc 1371 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))) ∈ (Base‘𝑅))
112 eqid 2736 . . . . . . . . 9 (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))
113 fvex 6855 . . . . . . . . . 10 (0g𝑅) ∈ V
114113a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (0g𝑅) ∈ V)
115112, 84, 111, 114fsuppmptdm 9316 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) finSupp (0g𝑅))
1162, 80, 81, 59, 82, 84, 93, 111, 115gsummulc1 20030 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))))) = ((𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))))(.r𝑅)(𝑍‘(𝑥f𝑘))))
11793adantr 481 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
1182, 59ringass 19984 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((𝑋𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅) ∧ (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
11994, 101, 109, 117, 118syl13anc 1372 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
1203psrbagf 21320 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐷𝑥:𝐼⟶ℕ0)
121120adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐷) → 𝑥:𝐼⟶ℕ0)
122121ad2antrr 724 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥:𝐼⟶ℕ0)
123122ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑥𝑧) ∈ ℕ0)
12488adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘:𝐼⟶ℕ0)
125124ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
126104ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
127 nn0cn 12423 . . . . . . . . . . . . . . . . 17 ((𝑥𝑧) ∈ ℕ0 → (𝑥𝑧) ∈ ℂ)
128 nn0cn 12423 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
129 nn0cn 12423 . . . . . . . . . . . . . . . . 17 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
130 nnncan2 11438 . . . . . . . . . . . . . . . . 17 (((𝑥𝑧) ∈ ℂ ∧ (𝑘𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
131127, 128, 129, 130syl3an 1160 . . . . . . . . . . . . . . . 16 (((𝑥𝑧) ∈ ℕ0 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
132123, 125, 126, 131syl3anc 1371 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
133132mpteq2dva 5205 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
134 psrring.i . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
135134ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝐼𝑉)
136135adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝐼𝑉)
137 ovexd 7392 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑥𝑧) − (𝑗𝑧)) ∈ V)
138 ovexd 7392 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑘𝑧) − (𝑗𝑧)) ∈ V)
139122feqmptd 6910 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥 = (𝑧𝐼 ↦ (𝑥𝑧)))
140104feqmptd 6910 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
141136, 123, 126, 139, 140offval2 7637 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑗) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑗𝑧))))
142124feqmptd 6910 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
143136, 125, 126, 142, 140offval2 7637 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑗𝑧))))
144136, 137, 138, 141, 143offval2 7637 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))))
145136, 123, 125, 139, 142offval2 7637 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑘) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
146133, 144, 1453eqtr4d 2786 . . . . . . . . . . . . 13 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑥f𝑘))
147146fveq2d 6846 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))) = (𝑍‘(𝑥f𝑘)))
148147oveq2d 7373 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘))))
149148oveq2d 7373 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
150119, 149eqtr4d 2779 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
151150mpteq2dva 5205 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))))
152151oveq2d 7373 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))
15379, 116, 1523eqtr2d 2782 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))
154153mpteq2dva 5205 . . . . 5 ((𝜑𝑥𝐷) → (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘)))) = (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))))))
155154oveq2d 7373 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))))
1568ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
15710ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍𝐵)
1581, 4, 59, 5, 3, 156, 157, 50psrmulval 21354 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑌 × 𝑍)‘(𝑥f𝑗)) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))
159158oveq2d 7373 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
1603psrbaglefi 21334 . . . . . . . . 9 ((𝑥f𝑗) ∈ 𝐷 → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
16150, 160syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
162 ovex 7390 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
1633, 162rab2ex 5292 . . . . . . . . . . . 12 {𝐷r ≤ (𝑥f𝑗)} ∈ V
164163mptex 7173 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V
165 funmpt 6539 . . . . . . . . . . 11 Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
166164, 165, 1133pm3.2i 1339 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V)
167166a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V))
168 suppssdm 8108 . . . . . . . . . . 11 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
169 eqid 2736 . . . . . . . . . . . 12 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
170169dmmptss 6193 . . . . . . . . . . 11 dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ⊆ {𝐷r ≤ (𝑥f𝑗)}
171168, 170sstri 3953 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)}
172171a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)})
173 suppssfifsupp 9320 . . . . . . . . 9 ((((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V) ∧ ({𝐷r ≤ (𝑥f𝑗)} ∈ Fin ∧ ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)})) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
174167, 161, 172, 173syl12anc 835 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
1752, 80, 81, 59, 23, 161, 32, 61, 174gsummulc2 20031 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
176159, 175eqtr4d 2779 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
177176mpteq2dva 5205 . . . . 5 ((𝜑𝑥𝐷) → (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗)))) = (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))))
178177oveq2d 7373 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))))
17970, 155, 1783eqtr4d 2786 . . 3 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
1809adantr 481 . . . 4 ((𝜑𝑥𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
18110adantr 481 . . . 4 ((𝜑𝑥𝐷) → 𝑍𝐵)
1821, 4, 59, 5, 3, 180, 181, 19psrmulval 21354 . . 3 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))))
1837adantr 481 . . . 4 ((𝜑𝑥𝐷) → 𝑋𝐵)
18414adantr 481 . . . 4 ((𝜑𝑥𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
1851, 4, 59, 5, 3, 183, 184, 19psrmulval 21354 . . 3 ((𝜑𝑥𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
186179, 182, 1853eqtr4d 2786 . 2 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
18713, 17, 186eqfnfvd 6985 1 (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  wss 3910   class class class wbr 5105  cmpt 5188  ccnv 5632  dom cdm 5633  cima 5636  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  r cofr 7616   supp csupp 8092  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  cc 11049  cle 11190  cmin 11385  cn 12153  0cn0 12413  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321   Σg cgsu 17322  CMndccmn 19562  Ringcrg 19964   mPwSer cmps 21306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-tset 17152  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-psr 21311
This theorem is referenced by:  psrring  21380
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