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| 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 22510 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )}) |
| 8 | 7 | eleq2d 2827 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})) |
| 9 | eqeq1 2741 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 ))) | |
| 10 | 9 | rexbidv 3179 | . . 3 ⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 ))) |
| 11 | 10 | elrab 3692 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 Basecbs 17247 ·𝑠 cvsca 17301 1rcur 20178 Mat cmat 22411 ScMat cscmat 22495 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-scmat 22497 |
| This theorem is referenced by: scmatscmid 22512 scmatmat 22515 scmatid 22520 scmataddcl 22522 scmatsubcl 22523 scmatmulcl 22524 smatvscl 22530 scmatrhmcl 22534 mat0scmat 22544 mat1scmat 22545 chmaidscmat 22854 |
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