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Theorem submaval 21930
Description: Third substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
Hypotheses
Ref Expression
submafval.a 𝐴 = (𝑁 Mat 𝑅)
submafval.q 𝑄 = (𝑁 subMat 𝑅)
submafval.b 𝐵 = (Base‘𝐴)
Assertion
Ref Expression
submaval ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝐾,𝑗   𝑖,𝐿,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖,𝑗)   𝑄(𝑖,𝑗)

Proof of Theorem submaval
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submafval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 submafval.q . . . 4 𝑄 = (𝑁 subMat 𝑅)
3 submafval.b . . . 4 𝐵 = (Base‘𝐴)
41, 2, 3submaval0 21929 . . 3 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
543ad2ant1 1133 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
6 simp2 1137 . . 3 ((𝑀𝐵𝐾𝑁𝐿𝑁) → 𝐾𝑁)
7 simpl3 1193 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
81, 3matrcl 21759 . . . . . . . . 9 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
98simpld 495 . . . . . . . 8 (𝑀𝐵𝑁 ∈ Fin)
10 diffi 9123 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑘}) ∈ Fin)
119, 10syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑘}) ∈ Fin)
12 diffi 9123 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑙}) ∈ Fin)
139, 12syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑙}) ∈ Fin)
1411, 13jca 512 . . . . . 6 (𝑀𝐵 → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
15143ad2ant1 1133 . . . . 5 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
1615adantr 481 . . . 4 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
17 mpoexga 8010 . . . 4 (((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) ∈ V)
1816, 17syl 17 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) ∈ V)
19 sneq 4596 . . . . . . 7 (𝑘 = 𝐾 → {𝑘} = {𝐾})
2019difeq2d 4082 . . . . . 6 (𝑘 = 𝐾 → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
2120adantr 481 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
22 sneq 4596 . . . . . . 7 (𝑙 = 𝐿 → {𝑙} = {𝐿})
2322difeq2d 4082 . . . . . 6 (𝑙 = 𝐿 → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
2423adantl 482 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
25 eqidd 2737 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖𝑀𝑗) = (𝑖𝑀𝑗))
2621, 24, 25mpoeq123dv 7432 . . . 4 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
2726adantl 482 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
286, 7, 18, 27ovmpodv2 7513 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗))))
295, 28mpd 15 1 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  {csn 4586  cfv 6496  (class class class)co 7357  cmpo 7359  Fincfn 8883  Basecbs 17083   Mat cmat 21754   subMat csubma 21925
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-en 8884  df-fin 8887  df-nn 12154  df-slot 17054  df-ndx 17066  df-base 17084  df-mat 21755  df-subma 21926
This theorem is referenced by:  submaeval  21931  1marepvsma1  21932  smadiadet  22019  submat1n  32386  submatres  32387  madjusmdetlem1  32408
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