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Theorem mdetunilem1 20237
 Description: Lemma for mdetuni 20247. (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 1062 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐹𝐺)
2 simpl3 1059 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))
3 simpr2 1061 . . 3 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐺𝑁)
4 simpl2 1058 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐸𝐵)
5 simpr1 1060 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝐹𝑁)
6 simpl1 1057 . . . . 5 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → 𝜑)
7 mdetuni.al . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
86, 7syl 17 . . . 4 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
9 oveq 6555 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑦𝑥𝑤) = (𝑦𝐸𝑤))
10 oveq 6555 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑧𝑥𝑤) = (𝑧𝐸𝑤))
119, 10eqeq12d 2625 . . . . . . . . 9 (𝑥 = 𝐸 → ((𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1211ralbidv 2969 . . . . . . . 8 (𝑥 = 𝐸 → (∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1312anbi2d 736 . . . . . . 7 (𝑥 = 𝐸 → ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) ↔ (𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤))))
14 fveq2 6103 . . . . . . . 8 (𝑥 = 𝐸 → (𝐷𝑥) = (𝐷𝐸))
1514eqeq1d 2612 . . . . . . 7 (𝑥 = 𝐸 → ((𝐷𝑥) = 0 ↔ (𝐷𝐸) = 0 ))
1613, 15imbi12d 333 . . . . . 6 (𝑥 = 𝐸 → (((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ) ↔ ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
1716ralbidv 2969 . . . . 5 (𝑥 = 𝐸 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ) ↔ ∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
18 neeq1 2844 . . . . . . . 8 (𝑦 = 𝐹 → (𝑦𝑧𝐹𝑧))
19 oveq1 6556 . . . . . . . . . 10 (𝑦 = 𝐹 → (𝑦𝐸𝑤) = (𝐹𝐸𝑤))
2019eqeq1d 2612 . . . . . . . . 9 (𝑦 = 𝐹 → ((𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
2120ralbidv 2969 . . . . . . . 8 (𝑦 = 𝐹 → (∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
2218, 21anbi12d 743 . . . . . . 7 (𝑦 = 𝐹 → ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤))))
2322imbi1d 330 . . . . . 6 (𝑦 = 𝐹 → (((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2423ralbidv 2969 . . . . 5 (𝑦 = 𝐹 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2517, 24rspc2va 3294 . . . 4 (((𝐸𝐵𝐹𝑁) ∧ ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 )) → ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ))
264, 5, 8, 25syl21anc 1317 . . 3 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ))
27 neeq2 2845 . . . . . 6 (𝑧 = 𝐺 → (𝐹𝑧𝐹𝐺))
28 oveq1 6556 . . . . . . . 8 (𝑧 = 𝐺 → (𝑧𝐸𝑤) = (𝐺𝐸𝑤))
2928eqeq2d 2620 . . . . . . 7 (𝑧 = 𝐺 → ((𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
3029ralbidv 2969 . . . . . 6 (𝑧 = 𝐺 → (∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
3127, 30anbi12d 743 . . . . 5 (𝑧 = 𝐺 → ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))))
3231imbi1d 330 . . . 4 (𝑧 = 𝐺 → (((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 )))
3332rspcva 3280 . . 3 ((𝐺𝑁 ∧ ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )) → ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 ))
343, 26, 33syl2anc 691 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 ))
351, 2, 34mp2and 711 1 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → (𝐷𝐸) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∖ cdif 3537  {csn 4125   × cxp 5036   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  Fincfn 7841  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  1rcur 18324  Ringcrg 18370   Mat cmat 20032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  mdetunilem2  20238  mdetuni0  20246
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