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Mirrors > Home > MPE Home > Th. List > mat1comp | Structured version Visualization version GIF version |
Description: The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.) |
Ref | Expression |
---|---|
mamumat1cl.b | ⊢ 𝐵 = (Base‘𝑅) |
mamumat1cl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mamumat1cl.o | ⊢ 1 = (1r‘𝑅) |
mamumat1cl.z | ⊢ 0 = (0g‘𝑅) |
mamumat1cl.i | ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) |
mamumat1cl.m | ⊢ (𝜑 → 𝑀 ∈ Fin) |
Ref | Expression |
---|---|
mat1comp | ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2730 | . . 3 ⊢ (𝑖 = 𝐴 → (𝑖 = 𝑗 ↔ 𝐴 = 𝑗)) | |
2 | 1 | ifbid 4546 | . 2 ⊢ (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 )) |
3 | eqeq2 2738 | . . 3 ⊢ (𝑗 = 𝐽 → (𝐴 = 𝑗 ↔ 𝐴 = 𝐽)) | |
4 | 3 | ifbid 4546 | . 2 ⊢ (𝑗 = 𝐽 → if(𝐴 = 𝑗, 1 , 0 ) = if(𝐴 = 𝐽, 1 , 0 )) |
5 | mamumat1cl.i | . 2 ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) | |
6 | mamumat1cl.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
7 | 6 | fvexi 6907 | . . 3 ⊢ 1 ∈ V |
8 | mamumat1cl.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
9 | 8 | fvexi 6907 | . . 3 ⊢ 0 ∈ V |
10 | 7, 9 | ifex 4573 | . 2 ⊢ if(𝐴 = 𝐽, 1 , 0 ) ∈ V |
11 | 2, 4, 5, 10 | ovmpo 7578 | 1 ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ifcif 4523 ‘cfv 6546 (class class class)co 7416 ∈ cmpo 7418 Fincfn 8966 Basecbs 17208 0gc0g 17449 1rcur 20160 Ringcrg 20212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 |
This theorem is referenced by: mamulid 22431 mamurid 22432 |
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