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Theorem psrass1 21280
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 21263 . . . . 5 (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10 psrass.z . . . . 5 (𝜑𝑍𝐵)
111, 4, 5, 6, 9, 10psrmulcl 21263 . . . 4 (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
121, 2, 3, 4, 11psrelbas 21254 . . 3 (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
1312ffnd 6652 . 2 (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
141, 4, 5, 6, 8, 10psrmulcl 21263 . . . . 5 (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
151, 4, 5, 6, 7, 14psrmulcl 21263 . . . 4 (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
161, 2, 3, 4, 15psrelbas 21254 . . 3 (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
1716ffnd 6652 . 2 (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18 eqid 2736 . . . . 5 {𝑔𝐷𝑔r𝑥} = {𝑔𝐷𝑔r𝑥}
19 simpr 485 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝐷)
20 ringcmn 19915 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
216, 20syl 17 . . . . . 6 (𝜑𝑅 ∈ CMnd)
2221adantr 481 . . . . 5 ((𝜑𝑥𝐷) → 𝑅 ∈ CMnd)
236ad2antrr 723 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
2423adantr 481 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑅 ∈ Ring)
251, 2, 3, 4, 7psrelbas 21254 . . . . . . . . . 10 (𝜑𝑋:𝐷⟶(Base‘𝑅))
2625ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
27 simpr 485 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗 ∈ {𝑔𝐷𝑔r𝑥})
28 breq1 5095 . . . . . . . . . . . 12 (𝑔 = 𝑗 → (𝑔r𝑥𝑗r𝑥))
2928elrab 3634 . . . . . . . . . . 11 (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑗𝐷𝑗r𝑥))
3027, 29sylib 217 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗𝐷𝑗r𝑥))
3130simpld 495 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗𝐷)
3226, 31ffvelcdmd 7018 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑋𝑗) ∈ (Base‘𝑅))
3332adantr 481 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑋𝑗) ∈ (Base‘𝑅))
341, 2, 3, 4, 8psrelbas 21254 . . . . . . . . . 10 (𝜑𝑌:𝐷⟶(Base‘𝑅))
3534ad3antrrr 727 . . . . . . . . 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 5095 . . . . . . . . . . . 12 ( = 𝑛 → (r ≤ (𝑥f𝑗) ↔ 𝑛r ≤ (𝑥f𝑗)))
3837elrab 3634 . . . . . . . . . . 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 7018 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑌𝑛) ∈ (Base‘𝑅))
421, 2, 3, 4, 10psrelbas 21254 . . . . . . . . . 10 (𝜑𝑍:𝐷⟶(Base‘𝑅))
4342ad3antrrr 727 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
44 simplr 766 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
453psrbagf 21227 . . . . . . . . . . . . . . 15 (𝑗𝐷𝑗:𝐼⟶ℕ0)
4631, 45syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗:𝐼⟶ℕ0)
4730simprd 496 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗r𝑥)
483psrbagcon 21239 . . . . . . . . . . . . . 14 ((𝑥𝐷𝑗:𝐼⟶ℕ0𝑗r𝑥) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
4944, 46, 47, 48syl3anc 1370 . . . . . . . . . . . . 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 21227 . . . . . . . . . . . 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 21239 . . . . . . . . . . 11 (((𝑥f𝑗) ∈ 𝐷𝑛:𝐼⟶ℕ0𝑛r ≤ (𝑥f𝑗)) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
5651, 53, 54, 55syl3anc 1370 . . . . . . . . . 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 7018 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅))
59 eqid 2736 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
602, 59ringcl 19895 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑌𝑛) ∈ (Base‘𝑅) ∧ (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅)) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
6124, 41, 58, 60syl3anc 1370 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
622, 59ringcl 19895 . . . . . . 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 1370 . . . . . 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 6825 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑌𝑛) = (𝑌‘(𝑘f𝑗)))
66 oveq2 7345 . . . . . . . 8 (𝑛 = (𝑘f𝑗) → ((𝑥f𝑗) ∘f𝑛) = ((𝑥f𝑗) ∘f − (𝑘f𝑗)))
6766fveq2d 6829 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) = (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))
6865, 67oveq12d 7355 . . . . . 6 (𝑛 = (𝑘f𝑗) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))
6968oveq2d 7353 . . . . 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 21252 . . . 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 723 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋𝐵)
728ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
73 simpr 485 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘 ∈ {𝑔𝐷𝑔r𝑥})
74 breq1 5095 . . . . . . . . . . . 12 (𝑔 = 𝑘 → (𝑔r𝑥𝑘r𝑥))
7574elrab 3634 . . . . . . . . . . 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 21261 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))))
7978oveq1d 7352 . . . . . . 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 723 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
833psrbaglefi 21241 . . . . . . . . 9 (𝑘𝐷 → {𝐷r𝑘} ∈ Fin)
8477, 83syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r𝑘} ∈ Fin)
8542ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
86 simplr 766 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
873psrbagf 21227 . . . . . . . . . . . 12 (𝑘𝐷𝑘:𝐼⟶ℕ0)
8877, 87syl 17 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘:𝐼⟶ℕ0)
8976simprd 496 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘r𝑥)
903psrbagcon 21239 . . . . . . . . . . 11 ((𝑥𝐷𝑘:𝐼⟶ℕ0𝑘r𝑥) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
9186, 88, 89, 90syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
9291simpld 495 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑘) ∈ 𝐷)
9385, 92ffvelcdmd 7018 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
9482adantr 481 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑅 ∈ Ring)
9525ad3antrrr 727 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
96 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 ∈ {𝐷r𝑘})
97 breq1 5095 . . . . . . . . . . . . 13 ( = 𝑗 → (r𝑘𝑗r𝑘))
9897elrab 3634 . . . . . . . . . . . 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 7018 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑋𝑗) ∈ (Base‘𝑅))
10234ad3antrrr 727 . . . . . . . . . 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 21239 . . . . . . . . . . . 12 ((𝑘𝐷𝑗:𝐼⟶ℕ0𝑗r𝑘) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
107103, 104, 105, 106syl3anc 1370 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
108107simpld 495 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
109102, 108ffvelcdmd 7018 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅))
1102, 59ringcl 19895 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑋𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅)) → ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))) ∈ (Base‘𝑅))
11194, 101, 109, 110syl3anc 1370 . . . . . . . 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 6838 . . . . . . . . . 10 (0g𝑅) ∈ V
114113a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (0g𝑅) ∈ V)
115112, 84, 111, 114fsuppmptdm 9237 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) finSupp (0g𝑅))
1162, 80, 81, 59, 82, 84, 93, 111, 115gsummulc1 19940 . . . . . . 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 19898 . . . . . . . . . . 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 1371 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
1203psrbagf 21227 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐷𝑥:𝐼⟶ℕ0)
121120adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐷) → 𝑥:𝐼⟶ℕ0)
122121ad2antrr 723 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥:𝐼⟶ℕ0)
123122ffvelcdmda 7017 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑥𝑧) ∈ ℕ0)
12488adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘:𝐼⟶ℕ0)
125124ffvelcdmda 7017 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
126104ffvelcdmda 7017 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
127 nn0cn 12344 . . . . . . . . . . . . . . . . 17 ((𝑥𝑧) ∈ ℕ0 → (𝑥𝑧) ∈ ℂ)
128 nn0cn 12344 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
129 nn0cn 12344 . . . . . . . . . . . . . . . . 17 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
130 nnncan2 11359 . . . . . . . . . . . . . . . . 17 (((𝑥𝑧) ∈ ℂ ∧ (𝑘𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
131127, 128, 129, 130syl3an 1159 . . . . . . . . . . . . . . . 16 (((𝑥𝑧) ∈ ℕ0 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
132123, 125, 126, 131syl3anc 1370 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
133132mpteq2dva 5192 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
134 psrring.i . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
135134ad2antrr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝐼𝑉)
136135adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝐼𝑉)
137 ovexd 7372 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑥𝑧) − (𝑗𝑧)) ∈ V)
138 ovexd 7372 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑘𝑧) − (𝑗𝑧)) ∈ V)
139122feqmptd 6893 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥 = (𝑧𝐼 ↦ (𝑥𝑧)))
140104feqmptd 6893 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
141136, 123, 126, 139, 140offval2 7615 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑗) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑗𝑧))))
142124feqmptd 6893 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
143136, 125, 126, 142, 140offval2 7615 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑗𝑧))))
144136, 137, 138, 141, 143offval2 7615 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))))
145136, 123, 125, 139, 142offval2 7615 . . . . . . . . . . . . . 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 6829 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))) = (𝑍‘(𝑥f𝑘)))
148147oveq2d 7353 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘))))
149148oveq2d 7353 . . . . . . . . . 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 5192 . . . . . . . 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 7353 . . . . . . 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 5192 . . . . 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 7353 . . . 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 723 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
15710ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍𝐵)
1581, 4, 59, 5, 3, 156, 157, 50psrmulval 21261 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑌 × 𝑍)‘(𝑥f𝑗)) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))
159158oveq2d 7353 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
1603psrbaglefi 21241 . . . . . . . . 9 ((𝑥f𝑗) ∈ 𝐷 → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
16150, 160syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
162 ovex 7370 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
1633, 162rab2ex 5279 . . . . . . . . . . . 12 {𝐷r ≤ (𝑥f𝑗)} ∈ V
164163mptex 7155 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V
165 funmpt 6522 . . . . . . . . . . 11 Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
166164, 165, 1133pm3.2i 1338 . . . . . . . . . 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 8063 . . . . . . . . . . 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 6179 . . . . . . . . . . 11 dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ⊆ {𝐷r ≤ (𝑥f𝑗)}
171168, 170sstri 3941 . . . . . . . . . 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 9241 . . . . . . . . 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 834 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
1752, 80, 81, 59, 23, 161, 32, 61, 174gsummulc2 19941 . . . . . . 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 5192 . . . . 5 ((𝜑𝑥𝐷) → (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗)))) = (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))))
178177oveq2d 7353 . . . 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 21261 . . 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 21261 . . 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 6968 1 (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  {crab 3403  Vcvv 3441  wss 3898   class class class wbr 5092  cmpt 5175  ccnv 5619  dom cdm 5620  cima 5623  Fun wfun 6473  wf 6475  cfv 6479  (class class class)co 7337  f cof 7593  r cofr 7594   supp csupp 8047  m cmap 8686  Fincfn 8804   finSupp cfsupp 9226  cc 10970  cle 11111  cmin 11306  cn 12074  0cn0 12334  Basecbs 17009  +gcplusg 17059  .rcmulr 17060  0gc0g 17247   Σg cgsu 17248  CMndccmn 19481  Ringcrg 19878   mPwSer cmps 21213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595  df-ofr 7596  df-om 7781  df-1st 7899  df-2nd 7900  df-supp 8048  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-map 8688  df-pm 8689  df-ixp 8757  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-fsupp 9227  df-oi 9367  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-z 12421  df-uz 12684  df-fz 13341  df-fzo 13484  df-seq 13823  df-hash 14146  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-mulr 17073  df-sca 17075  df-vsca 17076  df-tset 17078  df-0g 17249  df-gsum 17250  df-mre 17392  df-mrc 17393  df-acs 17395  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-mhm 18527  df-submnd 18528  df-grp 18676  df-minusg 18677  df-mulg 18797  df-ghm 18928  df-cntz 19019  df-cmn 19483  df-abl 19484  df-mgp 19816  df-ur 19833  df-ring 19880  df-psr 21218
This theorem is referenced by:  psrring  21286
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