![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > scmatel | Structured version Visualization version GIF version |
Description: An 𝑁 x 𝑁 scalar matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
Ref | Expression |
---|---|
scmatval.k | ⊢ 𝐾 = (Base‘𝑅) |
scmatval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
scmatval.b | ⊢ 𝐵 = (Base‘𝐴) |
scmatval.1 | ⊢ 1 = (1r‘𝐴) |
scmatval.t | ⊢ · = ( ·𝑠 ‘𝐴) |
scmatval.s | ⊢ 𝑆 = (𝑁 ScMat 𝑅) |
Ref | Expression |
---|---|
scmatel | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmatval.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
2 | scmatval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | scmatval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
4 | scmatval.1 | . . . 4 ⊢ 1 = (1r‘𝐴) | |
5 | scmatval.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐴) | |
6 | scmatval.s | . . . 4 ⊢ 𝑆 = (𝑁 ScMat 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | scmatval 22393 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )}) |
8 | 7 | eleq2d 2814 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})) |
9 | eqeq1 2731 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 ))) | |
10 | 9 | rexbidv 3173 | . . 3 ⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 ))) |
11 | 10 | elrab 3680 | . 2 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )} ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 ))) |
12 | 8, 11 | bitrdi 287 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 {crab 3427 ‘cfv 6542 (class class class)co 7414 Fincfn 8955 Basecbs 17171 ·𝑠 cvsca 17228 1rcur 20112 Mat cmat 22294 ScMat cscmat 22378 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-scmat 22380 |
This theorem is referenced by: scmatscmid 22395 scmatmat 22398 scmatid 22403 scmataddcl 22405 scmatsubcl 22406 scmatmulcl 22407 smatvscl 22413 scmatrhmcl 22417 mat0scmat 22427 mat1scmat 22428 chmaidscmat 22737 |
Copyright terms: Public domain | W3C validator |