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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 20310 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )}) |
8 | 7 | eleq2d 2687 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})) |
9 | eqeq1 2626 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 ))) | |
10 | 9 | rexbidv 3052 | . . 3 ⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 ))) |
11 | 10 | elrab 3363 | . 2 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )} ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 ))) |
12 | 8, 11 | syl6bb 276 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {crab 2916 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 Basecbs 15857 ·𝑠 cvsca 15945 1rcur 18501 Mat cmat 20213 ScMat cscmat 20295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-scmat 20297 |
This theorem is referenced by: scmatscmid 20312 scmatmat 20315 scmatid 20320 scmataddcl 20322 scmatsubcl 20323 scmatmulcl 20324 smatvscl 20330 scmatrhmcl 20334 mat0scmat 20344 mat1scmat 20345 chmaidscmat 20653 |
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