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Mirrors > Home > MPE Home > Th. List > submaval0 | Structured version Visualization version GIF version |
Description: Second substitution for a submatrix. (Contributed by AV, 28-Dec-2018.) |
Ref | Expression |
---|---|
submafval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
submafval.q | ⊢ 𝑄 = (𝑁 subMat 𝑅) |
submafval.b | ⊢ 𝐵 = (Base‘𝐴) |
Ref | Expression |
---|---|
submaval0 | ⊢ (𝑀 ∈ 𝐵 → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submafval.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | submafval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
3 | 1, 2 | matrcl 21021 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | 3 | simpld 497 | . . 3 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
5 | mpoexga 7775 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) ∈ V) | |
6 | 4, 4, 5 | syl2anc 586 | . 2 ⊢ (𝑀 ∈ 𝐵 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) ∈ V) |
7 | oveq 7162 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗)) | |
8 | 7 | mpoeq3dv 7233 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) |
9 | 8 | mpoeq3dv 7233 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)))) |
10 | submafval.q | . . . 4 ⊢ 𝑄 = (𝑁 subMat 𝑅) | |
11 | 1, 10, 2 | submafval 21188 | . . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) |
12 | 9, 11 | fvmptg 6766 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) ∈ V) → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)))) |
13 | 6, 12 | mpdan 685 | 1 ⊢ (𝑀 ∈ 𝐵 → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 {csn 4567 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Fincfn 8509 Basecbs 16483 Mat cmat 21016 subMat csubma 21185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-slot 16487 df-base 16489 df-mat 21017 df-subma 21186 |
This theorem is referenced by: submaval 21190 |
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