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Theorem cpmatmcllem 20743
Description: Lemma for cpmatmcl 20744. (Contributed by AV, 18-Nov-2019.)
Hypotheses
Ref Expression
cpmatsrngpmat.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmatsrngpmat.p 𝑃 = (Poly1𝑅)
cpmatsrngpmat.c 𝐶 = (𝑁 Mat 𝑃)
Assertion
Ref Expression
cpmatmcllem (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
Distinct variable groups:   𝐶,𝑖,𝑗   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝐶,𝑐   𝑁,𝑐,𝑥,𝑦,𝑖,𝑗   𝑃,𝑐   𝑅,𝑐,𝑥,𝑦   𝑦,𝑆   𝐶,𝑘   𝑘,𝑁,𝑐,𝑖,𝑗,𝑥,𝑦   𝑃,𝑘   𝑅,𝑘
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑖,𝑗)   𝑆(𝑥,𝑖,𝑗,𝑘,𝑐)

Proof of Theorem cpmatmcllem
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 cpmatsrngpmat.s . . . 4 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 cpmatsrngpmat.p . . . 4 𝑃 = (Poly1𝑅)
3 cpmatsrngpmat.c . . . 4 𝐶 = (𝑁 Mat 𝑃)
4 eqid 2771 . . . 4 (Base‘𝐶) = (Base‘𝐶)
51, 2, 3, 4cpmatelimp 20737 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅))))
61, 2, 3, 4cpmatelimp 20737 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
76adantr 466 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
8 ralcom 3246 . . . . . . . . . . . . . . . 16 (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ∀𝑗𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
9 r19.26-2 3213 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑙𝑁𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
10 ralcom 3246 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑙𝑁𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
119, 10bitr3i 266 . . . . . . . . . . . . . . . . . . . . . 22 ((∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
12 nfv 1995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁))
13 nfra1 3090 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
1412, 13nfan 1980 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑐((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
15 simp-4r 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
16 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Base‘𝑃) = (Base‘𝑃)
17 simplrl 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑖𝑁)
18 simpr 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
19 simplrl 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑥 ∈ (Base‘𝐶))
2019adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑥 ∈ (Base‘𝐶))
213, 16, 4, 17, 18, 20matecld 20449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22 simplrr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑗𝑁)
23 simplrr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑦 ∈ (Base‘𝐶))
2423adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑦 ∈ (Base‘𝐶))
253, 16, 4, 18, 22, 24matecld 20449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑘𝑦𝑗) ∈ (Base‘𝑃))
2615, 21, 25jca32 505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
2726adantlr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
28 oveq2 6801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
2928fveq2d 6336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
3029fveq1d 6334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
3130eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅)))
32 fvoveq1 6816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
3332fveq1d 6334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
3433eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
3531, 34anbi12d 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑙 = 𝑘 → ((((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
3635rspcva 3458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑘𝑁 ∧ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
3736a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ((𝑘𝑁 ∧ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
3837exp4b 417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑐 ∈ ℕ → (𝑘𝑁 → (∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))))
3938com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑘𝑁 → (𝑐 ∈ ℕ → (∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))))
4039imp31 404 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) ∧ 𝑐 ∈ ℕ) → (∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
4140ralimdva 3111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
4241impancom 439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (𝑘𝑁 → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
4342imp 393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
44 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (0g𝑅) = (0g𝑅)
45 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (.r𝑃) = (.r𝑃)
462, 16, 44, 45cply1mul 19879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))) → (∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅)))
4727, 43, 46sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) → ∀𝑐 ∈ ℕ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅))
4847r19.21bi 3081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) ∧ 𝑐 ∈ ℕ) → ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅))
4948an32s 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) ∧ 𝑘𝑁) → ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅))
5049mpteq2dva 4878 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐)) = (𝑘𝑁 ↦ (0g𝑅)))
5150oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))))
52 ringmnd 18764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
5352anim2i 603 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
5453ancomd 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
5544gsumz 17582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5654, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5756ad4antr 712 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5851, 57eqtrd 2805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅))
5958ex 397 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (𝑐 ∈ ℕ → (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
6014, 59ralrimi 3106 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅))
61 simp-4r 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
62 nnnn0 11501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
6362adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
642ply1ring 19833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
6564ad4antlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑃 ∈ Ring)
6616, 45ringcl 18769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6765, 21, 25, 66syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6867ralrimiva 3115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6968adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
70 simp-4l 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑁 ∈ Fin)
712, 16, 61, 63, 69, 70coe1fzgsumd 19887 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))))
7271eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → (((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅) ↔ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
7372ralbidva 3134 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
7473adantr 466 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
7560, 74mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
7675ex 397 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
7711, 76syl5bi 232 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
7877expd 400 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
7978expr 444 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (𝑗𝑁 → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
8079com23 86 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (𝑗𝑁 → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
8180imp31 404 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) ∧ 𝑗𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
8281ralimdva 3111 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) → (∀𝑗𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
838, 82syl5bi 232 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
8483ex 397 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
8584com23 86 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
8685impancom 439 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (𝑖𝑁 → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
8786imp 393 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
8887ralimdva 3111 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
8988ex 397 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9089expr 444 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐶) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
9190impd 396 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
927, 91syld 47 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦𝑆 → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9392com23 86 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9493ex 397 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ (Base‘𝐶) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
9594impd 396 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
965, 95syld 47 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9796imp32 405 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  cmpt 4863  cfv 6031  (class class class)co 6793  Fincfn 8109  cn 11222  0cn0 11494  Basecbs 16064  .rcmulr 16150  0gc0g 16308   Σg cgsu 16309  Mndcmnd 17502  Ringcrg 18755  Poly1cpl1 19762  coe1cco1 19763   Mat cmat 20430   ConstPolyMat ccpmat 20728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-ot 4325  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-ofr 7045  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-sup 8504  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-hom 16174  df-cco 16175  df-0g 16310  df-gsum 16311  df-prds 16316  df-pws 16318  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-grp 17633  df-minusg 17634  df-mulg 17749  df-subg 17799  df-ghm 17866  df-cntz 17957  df-cmn 18402  df-abl 18403  df-mgp 18698  df-ur 18710  df-ring 18757  df-subrg 18988  df-sra 19387  df-rgmod 19388  df-psr 19571  df-mpl 19573  df-opsr 19575  df-psr1 19765  df-ply1 19767  df-coe1 19768  df-dsmm 20293  df-frlm 20308  df-mat 20431  df-cpmat 20731
This theorem is referenced by:  cpmatmcl  20744
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