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Theorem mdetunilem1 21961
Description: Lemma for mdetuni 21971. (Contributed by SO, 14-Jul-2018.)
Hypotheses
Ref Expression
mdetuni.a 𝐴 = (𝑁 Mat 𝑅)
mdetuni.b 𝐵 = (Base‘𝐴)
mdetuni.k 𝐾 = (Base‘𝑅)
mdetuni.0g 0 = (0g𝑅)
mdetuni.1r 1 = (1r𝑅)
mdetuni.pg + = (+g𝑅)
mdetuni.tg · = (.r𝑅)
mdetuni.n (𝜑𝑁 ∈ Fin)
mdetuni.r (𝜑𝑅 ∈ Ring)
mdetuni.ff (𝜑𝐷:𝐵𝐾)
mdetuni.al (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
mdetuni.li (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
mdetuni.sc (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))
Assertion
Ref Expression
mdetunilem1 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → (𝐷𝐸) = 0 )
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧,𝑤   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐾,𝑦,𝑧,𝑤   𝑥,𝑁,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝑥, · ,𝑦,𝑧,𝑤   𝑥, + ,𝑦,𝑧,𝑤   𝑥, 0 ,𝑦,𝑧,𝑤   𝑥, 1 ,𝑦,𝑧,𝑤   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝐴,𝑦,𝑧,𝑤   𝑥,𝐸,𝑦,𝑧,𝑤   𝑥,𝐹,𝑦,𝑧,𝑤   𝑥,𝐺,𝑦,𝑧,𝑤

Proof of Theorem mdetunilem1
StepHypRef Expression
1 simpr3 1196 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐹𝐺)
2 simpl3 1193 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))
3 neeq2 3007 . . . . 5 (𝑧 = 𝐺 → (𝐹𝑧𝐹𝐺))
4 oveq1 7364 . . . . . . 7 (𝑧 = 𝐺 → (𝑧𝐸𝑤) = (𝐺𝐸𝑤))
54eqeq2d 2747 . . . . . 6 (𝑧 = 𝐺 → ((𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
65ralbidv 3174 . . . . 5 (𝑧 = 𝐺 → (∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
73, 6anbi12d 631 . . . 4 (𝑧 = 𝐺 → ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))))
87imbi1d 341 . . 3 (𝑧 = 𝐺 → (((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 )))
9 simpl2 1192 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐸𝐵)
10 simpr1 1194 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐹𝑁)
11 simpl1 1191 . . . . 5 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝜑)
12 mdetuni.al . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
1311, 12syl 17 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
14 oveq 7363 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑦𝑥𝑤) = (𝑦𝐸𝑤))
15 oveq 7363 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑧𝑥𝑤) = (𝑧𝐸𝑤))
1614, 15eqeq12d 2752 . . . . . . . . 9 (𝑥 = 𝐸 → ((𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1716ralbidv 3174 . . . . . . . 8 (𝑥 = 𝐸 → (∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1817anbi2d 629 . . . . . . 7 (𝑥 = 𝐸 → ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) ↔ (𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤))))
19 fveqeq2 6851 . . . . . . 7 (𝑥 = 𝐸 → ((𝐷𝑥) = 0 ↔ (𝐷𝐸) = 0 ))
2018, 19imbi12d 344 . . . . . 6 (𝑥 = 𝐸 → (((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ) ↔ ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2120ralbidv 3174 . . . . 5 (𝑥 = 𝐸 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ) ↔ ∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
22 neeq1 3006 . . . . . . . 8 (𝑦 = 𝐹 → (𝑦𝑧𝐹𝑧))
23 oveq1 7364 . . . . . . . . . 10 (𝑦 = 𝐹 → (𝑦𝐸𝑤) = (𝐹𝐸𝑤))
2423eqeq1d 2738 . . . . . . . . 9 (𝑦 = 𝐹 → ((𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
2524ralbidv 3174 . . . . . . . 8 (𝑦 = 𝐹 → (∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
2622, 25anbi12d 631 . . . . . . 7 (𝑦 = 𝐹 → ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤))))
2726imbi1d 341 . . . . . 6 (𝑦 = 𝐹 → (((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2827ralbidv 3174 . . . . 5 (𝑦 = 𝐹 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2921, 28rspc2va 3591 . . . 4 (((𝐸𝐵𝐹𝑁) ∧ ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 )) → ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ))
309, 10, 13, 29syl21anc 836 . . 3 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ))
31 simpr2 1195 . . 3 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐺𝑁)
328, 30, 31rspcdva 3582 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 ))
331, 2, 32mp2and 697 1 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → (𝐷𝐸) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  cdif 3907  {csn 4586   × cxp 5631  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  Fincfn 8883  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321  1rcur 19913  Ringcrg 19964   Mat cmat 21754
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-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360
This theorem is referenced by:  mdetunilem2  21962  mdetuni0  21970
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