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Mirrors > Home > MPE Home > Th. List > matbas | Structured version Visualization version GIF version |
Description: The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
matbas.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matbas.g | ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) |
Ref | Expression |
---|---|
matbas | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matbas.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | matbas.g | . . . 4 ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) | |
3 | eqid 2821 | . . . 4 ⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
4 | 1, 2, 3 | matval 20950 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉)) |
5 | 4 | fveq2d 6668 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘𝐴) = (Base‘(𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉))) |
6 | baseid 16533 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
7 | basendxnmulrndx 16608 | . . 3 ⊢ (Base‘ndx) ≠ (.r‘ndx) | |
8 | 6, 7 | setsnid 16529 | . 2 ⊢ (Base‘𝐺) = (Base‘(𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉)) |
9 | 5, 8 | syl6reqr 2875 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4565 〈cotp 4567 × cxp 5547 ‘cfv 6349 (class class class)co 7145 Fincfn 8498 ndxcnx 16470 sSet csts 16471 Basecbs 16473 .rcmulr 16556 freeLMod cfrlm 20820 maMul cmmul 20924 Mat cmat 20946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-mulr 16569 df-mat 20947 |
This theorem is referenced by: mat0 20956 matinvg 20957 matbas2 20960 matplusg2 20966 matvsca2 20967 matlmod 20968 matsubg 20971 matsubgcell 20973 matgsum 20976 matdim 30913 matunitlindflem2 34771 matunitlindf 34772 |
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