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| Mirrors > Home > MPE Home > Th. List > marepveval | Structured version Visualization version GIF version | ||
| Description: An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
| Ref | Expression |
|---|---|
| marepvfval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| marepvfval.b | ⊢ 𝐵 = (Base‘𝐴) |
| marepvfval.q | ⊢ 𝑄 = (𝑁 matRepV 𝑅) |
| marepvfval.v | ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) |
| Ref | Expression |
|---|---|
| marepveval | ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼𝑀𝐽))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | marepvfval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 2 | marepvfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
| 3 | marepvfval.q | . . . 4 ⊢ 𝑄 = (𝑁 matRepV 𝑅) | |
| 4 | marepvfval.v | . . . 4 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) | |
| 5 | 1, 2, 3, 4 | marepvval 22573 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)))) |
| 6 | 5 | adantr 480 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)))) |
| 7 | simprl 771 | . . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁) | |
| 8 | simplrr 778 | . . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑖 = 𝐼) → 𝐽 ∈ 𝑁) | |
| 9 | fvexd 6921 | . . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐶‘𝑖) ∈ V) | |
| 10 | ovexd 7466 | . . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ V) | |
| 11 | 9, 10 | ifcld 4572 | . . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)) ∈ V) |
| 12 | 11 | adantr 480 | . . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)) ∈ V) |
| 13 | eqeq1 2741 | . . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑗 = 𝐾 ↔ 𝐽 = 𝐾)) | |
| 14 | 13 | adantl 481 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑗 = 𝐾 ↔ 𝐽 = 𝐾)) |
| 15 | fveq2 6906 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (𝐶‘𝑖) = (𝐶‘𝐼)) | |
| 16 | 15 | adantr 480 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝐶‘𝑖) = (𝐶‘𝐼)) |
| 17 | oveq12 7440 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽)) | |
| 18 | 14, 16, 17 | ifbieq12d 4554 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼𝑀𝐽))) |
| 19 | 18 | adantl 481 | . . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼𝑀𝐽))) |
| 20 | 7, 8, 12, 19 | ovmpodv2 7591 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (((𝑀𝑄𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼𝑀𝐽)))) |
| 21 | 6, 20 | mpd 15 | 1 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼𝑀𝐽))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ifcif 4525 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8866 Basecbs 17247 Mat cmat 22411 matRepV cmatrepV 22563 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-mat 22412 df-marepv 22565 |
| This theorem is referenced by: ma1repveval 22577 1marepvsma1 22589 |
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