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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 20299 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))) |
6 | simpl 473 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → 𝑀 ∈ 𝐵) | |
7 | 5, 6 | syl6bi 243 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 Basecbs 15857 0gc0g 16100 Mat cmat 20213 DMat cdmat 20294 |
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-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-dmat 20296 |
This theorem is referenced by: dmatmul 20303 dmatsubcl 20304 dmatsgrp 20305 dmatmulcl 20306 dmatcrng 20308 dmatscmcl 20309 |
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