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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 2610 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | scmatmat.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | scmatmat.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | eqid 2610 | . . 3 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
5 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴) | |
6 | scmatmat.s | . . 3 ⊢ 𝑆 = (𝑁 ScMat 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | scmatel 20130 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))))) |
8 | simpl 472 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))) → 𝑀 ∈ 𝐵) | |
9 | 7, 8 | syl6bi 242 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 ·𝑠 cvsca 15772 1rcur 18324 Mat cmat 20032 ScMat cscmat 20114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-scmat 20116 |
This theorem is referenced by: scmatsgrp 20144 scmatcrng 20146 |
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