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Theorem submaval 22603
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 22602 . . 3 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
543ad2ant1 1132 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
6 simp2 1136 . . 3 ((𝑀𝐵𝐾𝑁𝐿𝑁) → 𝐾𝑁)
7 simpl3 1192 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
81, 3matrcl 22432 . . . . . . . . 9 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
98simpld 494 . . . . . . . 8 (𝑀𝐵𝑁 ∈ Fin)
10 diffi 9214 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑘}) ∈ Fin)
119, 10syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑘}) ∈ Fin)
12 diffi 9214 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑙}) ∈ Fin)
139, 12syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑙}) ∈ Fin)
1411, 13jca 511 . . . . . 6 (𝑀𝐵 → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
15143ad2ant1 1132 . . . . 5 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
1615adantr 480 . . . 4 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
17 mpoexga 8101 . . . 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 4641 . . . . . . 7 (𝑘 = 𝐾 → {𝑘} = {𝐾})
2019difeq2d 4136 . . . . . 6 (𝑘 = 𝐾 → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
2120adantr 480 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
22 sneq 4641 . . . . . . 7 (𝑙 = 𝐿 → {𝑙} = {𝐿})
2322difeq2d 4136 . . . . . 6 (𝑙 = 𝐿 → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
2423adantl 481 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
25 eqidd 2736 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖𝑀𝑗) = (𝑖𝑀𝑗))
2621, 24, 25mpoeq123dv 7508 . . . 4 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
2726adantl 481 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
286, 7, 18, 27ovmpodv2 7591 . 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 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  {csn 4631  cfv 6563  (class class class)co 7431  cmpo 7433  Fincfn 8984  Basecbs 17245   Mat cmat 22427   subMat csubma 22598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-en 8985  df-fin 8988  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-mat 22428  df-subma 22599
This theorem is referenced by:  submaeval  22604  1marepvsma1  22605  smadiadet  22692  submat1n  33766  submatres  33767  madjusmdetlem1  33788
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