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Theorem mdetunilem1 20941
 Description: Lemma for mdetuni 20951. (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 1177 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐹𝐺)
2 simpl3 1174 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))
3 neeq2 3025 . . . . 5 (𝑧 = 𝐺 → (𝐹𝑧𝐹𝐺))
4 oveq1 6982 . . . . . . 7 (𝑧 = 𝐺 → (𝑧𝐸𝑤) = (𝐺𝐸𝑤))
54eqeq2d 2783 . . . . . 6 (𝑧 = 𝐺 → ((𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
65ralbidv 3142 . . . . 5 (𝑧 = 𝐺 → (∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
73, 6anbi12d 622 . . . 4 (𝑧 = 𝐺 → ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))))
87imbi1d 334 . . 3 (𝑧 = 𝐺 → (((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 )))
9 simpl2 1173 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐸𝐵)
10 simpr1 1175 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐹𝑁)
11 simpl1 1172 . . . . 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 6981 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑦𝑥𝑤) = (𝑦𝐸𝑤))
15 oveq 6981 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑧𝑥𝑤) = (𝑧𝐸𝑤))
1614, 15eqeq12d 2788 . . . . . . . . 9 (𝑥 = 𝐸 → ((𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1716ralbidv 3142 . . . . . . . 8 (𝑥 = 𝐸 → (∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1817anbi2d 620 . . . . . . 7 (𝑥 = 𝐸 → ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) ↔ (𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤))))
19 fveqeq2 6506 . . . . . . 7 (𝑥 = 𝐸 → ((𝐷𝑥) = 0 ↔ (𝐷𝐸) = 0 ))
2018, 19imbi12d 337 . . . . . 6 (𝑥 = 𝐸 → (((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ) ↔ ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2120ralbidv 3142 . . . . 5 (𝑥 = 𝐸 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ) ↔ ∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
22 neeq1 3024 . . . . . . . 8 (𝑦 = 𝐹 → (𝑦𝑧𝐹𝑧))
23 oveq1 6982 . . . . . . . . . 10 (𝑦 = 𝐹 → (𝑦𝐸𝑤) = (𝐹𝐸𝑤))
2423eqeq1d 2775 . . . . . . . . 9 (𝑦 = 𝐹 → ((𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
2524ralbidv 3142 . . . . . . . 8 (𝑦 = 𝐹 → (∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
2622, 25anbi12d 622 . . . . . . 7 (𝑦 = 𝐹 → ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤))))
2726imbi1d 334 . . . . . 6 (𝑦 = 𝐹 → (((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2827ralbidv 3142 . . . . 5 (𝑦 = 𝐹 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2921, 28rspc2va 3544 . . . 4 (((𝐸𝐵𝐹𝑁) ∧ ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 )) → ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ))
309, 10, 13, 29syl21anc 826 . . 3 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ))
31 simpr2 1176 . . 3 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐺𝑁)
328, 30, 31rspcdva 3536 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 ))
331, 2, 32mp2and 687 1 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → (𝐷𝐸) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ≠ wne 2962  ∀wral 3083   ∖ cdif 3821  {csn 4436   × cxp 5402   ↾ cres 5406  ⟶wf 6182  ‘cfv 6186  (class class class)co 6975   ∘𝑓 cof 7224  Fincfn 8305  Basecbs 16338  +gcplusg 16420  .rcmulr 16421  0gc0g 16568  1rcur 18987  Ringcrg 19033   Mat cmat 20736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-iota 6150  df-fv 6194  df-ov 6978 This theorem is referenced by:  mdetunilem2  20942  mdetuni0  20950
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