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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 2732 | . . 3 ⊢ (𝑖 = 𝐴 → (𝑖 = 𝑗 ↔ 𝐴 = 𝑗)) | |
2 | 1 | ifbid 4555 | . 2 ⊢ (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 )) |
3 | eqeq2 2740 | . . 3 ⊢ (𝑗 = 𝐽 → (𝐴 = 𝑗 ↔ 𝐴 = 𝐽)) | |
4 | 3 | ifbid 4555 | . 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 6916 | . . 3 ⊢ 1 ∈ V |
8 | mamumat1cl.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
9 | 8 | fvexi 6916 | . . 3 ⊢ 0 ∈ V |
10 | 7, 9 | ifex 4582 | . 2 ⊢ if(𝐴 = 𝐽, 1 , 0 ) ∈ V |
11 | 2, 4, 5, 10 | ovmpo 7587 | 1 ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ifcif 4532 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 Fincfn 8970 Basecbs 17187 0gc0g 17428 1rcur 20128 Ringcrg 20180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 |
This theorem is referenced by: mamulid 22363 mamurid 22364 |
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