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Mirrors > Home > MPE Home > Th. List > dmatmat | Structured version Visualization version GIF version |
Description: An 𝑁 x 𝑁 diagonal matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
Ref | Expression |
---|---|
dmatval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
dmatval.b | ⊢ 𝐵 = (Base‘𝐴) |
dmatval.0 | ⊢ 0 = (0g‘𝑅) |
dmatval.d | ⊢ 𝐷 = (𝑁 DMat 𝑅) |
Ref | Expression |
---|---|
dmatmat | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmatval.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | dmatval.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
3 | dmatval.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | dmatval.d | . . 3 ⊢ 𝐷 = (𝑁 DMat 𝑅) | |
5 | 1, 2, 3, 4 | dmatel 21986 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))) |
6 | simpl 483 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → 𝑀 ∈ 𝐵) | |
7 | 5, 6 | syl6bi 252 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 Basecbs 17140 0gc0g 17381 Mat cmat 21898 DMat cdmat 21981 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-dmat 21983 |
This theorem is referenced by: dmatmul 21990 dmatsubcl 21991 dmatsgrp 21992 dmatmulcl 21993 dmatcrng 21995 dmatscmcl 21996 |
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