MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mdetunilem1 Structured version   Visualization version   GIF version

Theorem mdetunilem1 22639
Description: Lemma for mdetuni 22649. (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 3010 . . . . 5 (𝑧 = 𝐺 → (𝐹𝑧𝐹𝐺))
4 oveq1 7455 . . . . . . 7 (𝑧 = 𝐺 → (𝑧𝐸𝑤) = (𝐺𝐸𝑤))
54eqeq2d 2751 . . . . . 6 (𝑧 = 𝐺 → ((𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
65ralbidv 3184 . . . . 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 7454 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑦𝑥𝑤) = (𝑦𝐸𝑤))
15 oveq 7454 . . . . . . . . . 10 (𝑥 = 𝐸 → (𝑧𝑥𝑤) = (𝑧𝐸𝑤))
1614, 15eqeq12d 2756 . . . . . . . . 9 (𝑥 = 𝐸 → ((𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1716ralbidv 3184 . . . . . . . 8 (𝑥 = 𝐸 → (∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
1817anbi2d 629 . . . . . . 7 (𝑥 = 𝐸 → ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) ↔ (𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤))))
19 fveqeq2 6929 . . . . . . 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 3184 . . . . 5 (𝑥 = 𝐸 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ) ↔ ∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
22 neeq1 3009 . . . . . . . 8 (𝑦 = 𝐹 → (𝑦𝑧𝐹𝑧))
23 oveq1 7455 . . . . . . . . . 10 (𝑦 = 𝐹 → (𝑦𝐸𝑤) = (𝐹𝐸𝑤))
2423eqeq1d 2742 . . . . . . . . 9 (𝑦 = 𝐹 → ((𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
2524ralbidv 3184 . . . . . . . 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 3184 . . . . 5 (𝑦 = 𝐹 → (∀𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ) ↔ ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 )))
2921, 28rspc2va 3647 . . . 4 (((𝐸𝐵𝐹𝑁) ∧ ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 )) → ∀𝑧𝑁 ((𝐹𝑧 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷𝐸) = 0 ))
309, 10, 13, 29syl21anc 837 . . 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 3636 . 2 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → ((𝐹𝐺 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷𝐸) = 0 ))
331, 2, 32mp2and 698 1 (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → (𝐷𝐸) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wral 3067  cdif 3973  {csn 4648   × cxp 5698  cres 5702  wf 6569  cfv 6573  (class class class)co 7448  f cof 7712  Fincfn 9003  Basecbs 17258  +gcplusg 17311  .rcmulr 17312  0gc0g 17499  1rcur 20208  Ringcrg 20260   Mat cmat 22432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451
This theorem is referenced by:  mdetunilem2  22640  mdetuni0  22648
  Copyright terms: Public domain W3C validator