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Theorem submaval 21332
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 21331 . . 3 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
543ad2ant1 1134 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
6 simp2 1138 . . 3 ((𝑀𝐵𝐾𝑁𝐿𝑁) → 𝐾𝑁)
7 simpl3 1194 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
81, 3matrcl 21163 . . . . . . . . 9 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
98simpld 498 . . . . . . . 8 (𝑀𝐵𝑁 ∈ Fin)
10 diffi 8827 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑘}) ∈ Fin)
119, 10syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑘}) ∈ Fin)
12 diffi 8827 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑙}) ∈ Fin)
139, 12syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑙}) ∈ Fin)
1411, 13jca 515 . . . . . 6 (𝑀𝐵 → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
15143ad2ant1 1134 . . . . 5 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
1615adantr 484 . . . 4 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
17 mpoexga 7801 . . . 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 4526 . . . . . . 7 (𝑘 = 𝐾 → {𝑘} = {𝐾})
2019difeq2d 4013 . . . . . 6 (𝑘 = 𝐾 → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
2120adantr 484 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
22 sneq 4526 . . . . . . 7 (𝑙 = 𝐿 → {𝑙} = {𝐿})
2322difeq2d 4013 . . . . . 6 (𝑙 = 𝐿 → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
2423adantl 485 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
25 eqidd 2739 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖𝑀𝑗) = (𝑖𝑀𝑗))
2621, 24, 25mpoeq123dv 7243 . . . 4 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
2726adantl 485 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
286, 7, 18, 27ovmpodv2 7323 . 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 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3398  cdif 3840  {csn 4516  cfv 6339  (class class class)co 7170  cmpo 7172  Fincfn 8555  Basecbs 16586   Mat cmat 21158   subMat csubma 21327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-1o 8131  df-en 8556  df-fin 8559  df-slot 16590  df-base 16592  df-mat 21159  df-subma 21328
This theorem is referenced by:  submaeval  21333  1marepvsma1  21334  smadiadet  21421  submat1n  31327  submatres  31328  madjusmdetlem1  31349
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