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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 21102 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))) |
6 | simpl 485 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → 𝑀 ∈ 𝐵) | |
7 | 5, 6 | syl6bi 255 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 Basecbs 16483 0gc0g 16713 Mat cmat 21016 DMat cdmat 21097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-dmat 21099 |
This theorem is referenced by: dmatmul 21106 dmatsubcl 21107 dmatsgrp 21108 dmatmulcl 21109 dmatcrng 21111 dmatscmcl 21112 |
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