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Mirrors > Home > MPE Home > Th. List > marrepval0 | Structured version Visualization version GIF version |
Description: Second substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) |
Ref | Expression |
---|---|
marrepfval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
marrepfval.b | ⊢ 𝐵 = (Base‘𝐴) |
marrepfval.q | ⊢ 𝑄 = (𝑁 matRRep 𝑅) |
marrepfval.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
marrepval0 | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marrepfval.a | . . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | marrepfval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐴) | |
3 | 1, 2 | matrcl 20037 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | 3 | simpld 474 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
5 | 4, 4 | jca 553 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) |
7 | mpt2exga 7135 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V) | |
8 | 6, 7 | syl 17 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V) |
9 | ifeq1 4040 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 )) | |
10 | 9 | adantl 481 | . . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 )) |
11 | oveq 6555 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗)) | |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗)) |
13 | 10, 12 | ifeq12d 4056 | . . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) |
14 | 13 | mpt2eq3dv 6619 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) |
15 | 14 | mpt2eq3dv 6619 | . . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))) |
16 | marrepfval.q | . . . 4 ⊢ 𝑄 = (𝑁 matRRep 𝑅) | |
17 | marrepfval.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
18 | 1, 2, 16, 17 | marrepfval 20185 | . . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) |
19 | 15, 18 | ovmpt2ga 6688 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))) |
20 | 8, 19 | mpd3an3 1417 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Fincfn 7841 Basecbs 15695 0gc0g 15923 Mat cmat 20032 matRRep cmarrep 20181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-slot 15699 df-base 15700 df-mat 20033 df-marrep 20183 |
This theorem is referenced by: marrepval 20187 minmar1marrep 20275 |
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