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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 2802 | . . 3 ⊢ (𝑖 = 𝐴 → (𝑖 = 𝑗 ↔ 𝐴 = 𝑗)) | |
2 | 1 | ifbid 4447 | . 2 ⊢ (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 )) |
3 | eqeq2 2810 | . . 3 ⊢ (𝑗 = 𝐽 → (𝐴 = 𝑗 ↔ 𝐴 = 𝐽)) | |
4 | 3 | ifbid 4447 | . 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 6659 | . . 3 ⊢ 1 ∈ V |
8 | mamumat1cl.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
9 | 8 | fvexi 6659 | . . 3 ⊢ 0 ∈ V |
10 | 7, 9 | ifex 4473 | . 2 ⊢ if(𝐴 = 𝐽, 1 , 0 ) ∈ V |
11 | 2, 4, 5, 10 | ovmpo 7289 | 1 ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ifcif 4425 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Fincfn 8492 Basecbs 16475 0gc0g 16705 1rcur 19244 Ringcrg 19290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: mamulid 21046 mamurid 21047 |
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