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Theorem cpmatmcllem 20930
Description: Lemma for cpmatmcl 20931. (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 2777 . . . 4 (Base‘𝐶) = (Base‘𝐶)
51, 2, 3, 4cpmatelimp 20924 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅))))
61, 2, 3, 4cpmatelimp 20924 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
76adantr 474 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
8 ralcom 3283 . . . . . . . . . . . . . . . 16 (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ∀𝑗𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
9 r19.26-2 3250 . . . . . . . . . . . . . . . . . . . . . . 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 3283 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑙𝑁𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
119, 10bitr3i 269 . . . . . . . . . . . . . . . . . . . . . 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 1957 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁))
13 nfra1 3122 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
1412, 13nfan 1946 . . . . . . . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
16 eqid 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Base‘𝑃) = (Base‘𝑃)
17 simplrl 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑖𝑁)
18 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
19 simplrl 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑥 ∈ (Base‘𝐶))
2019adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑥 ∈ (Base‘𝐶))
213, 16, 4, 17, 18, 20matecld 20636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22 simplrr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑗𝑁)
23 simplrr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑦 ∈ (Base‘𝐶))
2423adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑦 ∈ (Base‘𝐶))
253, 16, 4, 18, 22, 24matecld 20636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑘𝑦𝑗) ∈ (Base‘𝑃))
2615, 21, 25jca32 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
2726adantlr 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6930 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
2928fveq2d 6450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
3029fveq1d 6448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
3130eqeq1d 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅)))
32 fvoveq1 6945 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
3332fveq1d 6448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
3433eqeq1d 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
3531, 34anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑙 = 𝑘 → ((((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
3635rspcva 3508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (0g𝑅) = (0g𝑅)
45 eqid 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (.r𝑃) = (.r𝑃)
462, 16, 44, 45cply1mul 20060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4979 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6938 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
5352anim2i 610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
5453ancomd 455 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
5544gsumz 17760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5654, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5756ad4antr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . . . . . . . . . . . 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 3138 . . . . . . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
62 nnnn0 11650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
6362adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
642ply1ring 20014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
6564ad4antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑃 ∈ Ring)
6616, 45ringcl 18948 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6765, 21, 25, 66syl3anc 1439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6867ralrimiva 3147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6968adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
70 simp-4l 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑁 ∈ Fin)
712, 16, 61, 63, 69, 70coe1fzgsumd 20068 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . 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 445 . . . . . . . . . . . 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 397 . . . . . . . . . . 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 3143 . . . . . . . . . 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 403 . . . . . . . . 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 450 . . . . . . . 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 400 . . . . . . 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 403 . . . 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 400 . . 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 411 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 198  wa 386   = wceq 1601  wcel 2106  wral 3089  cmpt 4965  cfv 6135  (class class class)co 6922  Fincfn 8241  cn 11374  0cn0 11642  Basecbs 16255  .rcmulr 16339  0gc0g 16486   Σg cgsu 16487  Mndcmnd 17680  Ringcrg 18934  Poly1cpl1 19943  coe1cco1 19944   Mat cmat 20617   ConstPolyMat ccpmat 20915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-ot 4406  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-ofr 7175  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-sup 8636  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-ip 16356  df-tset 16357  df-ple 16358  df-ds 16360  df-hom 16362  df-cco 16363  df-0g 16488  df-gsum 16489  df-prds 16494  df-pws 16496  df-mre 16632  df-mrc 16633  df-acs 16635  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-mhm 17721  df-submnd 17722  df-grp 17812  df-minusg 17813  df-mulg 17928  df-subg 17975  df-ghm 18042  df-cntz 18133  df-cmn 18581  df-abl 18582  df-mgp 18877  df-ur 18889  df-ring 18936  df-subrg 19170  df-sra 19569  df-rgmod 19570  df-psr 19753  df-mpl 19755  df-opsr 19757  df-psr1 19946  df-ply1 19948  df-coe1 19949  df-dsmm 20475  df-frlm 20490  df-mat 20618  df-cpmat 20918
This theorem is referenced by:  cpmatmcl  20931
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