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Mirrors > Home > MPE Home > Th. List > scmatmat | Structured version Visualization version GIF version |
Description: An 𝑁 x 𝑁 scalar matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
Ref | Expression |
---|---|
scmatmat.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
scmatmat.b | ⊢ 𝐵 = (Base‘𝐴) |
scmatmat.s | ⊢ 𝑆 = (𝑁 ScMat 𝑅) |
Ref | Expression |
---|---|
scmatmat | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | scmatmat.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | scmatmat.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | eqid 2736 | . . 3 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
5 | eqid 2736 | . . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴) | |
6 | scmatmat.s | . . 3 ⊢ 𝑆 = (𝑁 ScMat 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | scmatel 21854 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))))) |
8 | simpl 483 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))) → 𝑀 ∈ 𝐵) | |
9 | 7, 8 | syl6bi 252 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 Basecbs 17083 ·𝑠 cvsca 17137 1rcur 19913 Mat cmat 21754 ScMat cscmat 21838 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-scmat 21840 |
This theorem is referenced by: scmatsgrp 21868 scmatcrng 21870 |
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