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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dmatbas | Structured version Visualization version GIF version | ||
| Description: The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.) |
| Ref | Expression |
|---|---|
| dmatbas.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| dmatbas.b | ⊢ 𝐵 = (Base‘𝐴) |
| dmatbas.0 | ⊢ 0 = (0g‘𝑅) |
| dmatbas.d | ⊢ 𝐷 = (𝑁 DMat 𝑅) |
| Ref | Expression |
|---|---|
| dmatbas | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmatbas.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 2 | dmatbas.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
| 3 | dmatbas.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
| 4 | dmatbas.d | . . 3 ⊢ 𝐷 = (𝑁 DMat 𝑅) | |
| 5 | 1, 2, 3, 4 | dmatval 22498 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) |
| 6 | elex 3501 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
| 7 | eqid 2737 | . . . 4 ⊢ (𝑁 DMatALT 𝑅) = (𝑁 DMatALT 𝑅) | |
| 8 | 1, 2, 3, 7 | dmatALTbas 48318 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) |
| 9 | 6, 8 | sylan2 593 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) |
| 10 | 5, 9 | eqtr4d 2780 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 Basecbs 17247 0gc0g 17484 Mat cmat 22411 DMat cdmat 22494 DMatALT cdmatalt 48313 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-dmat 22496 df-dmatalt 48315 |
| This theorem is referenced by: (None) |
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