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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 | baseid 16913 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
2 | basendxnmulrndx 17003 | . . 3 ⊢ (Base‘ndx) ≠ (.r‘ndx) | |
3 | 1, 2 | setsnid 16908 | . 2 ⊢ (Base‘𝐺) = (Base‘(𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉)) |
4 | matbas.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | matbas.g | . . . 4 ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) | |
6 | eqid 2740 | . . . 4 ⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
7 | 4, 5, 6 | matval 21556 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉)) |
8 | 7 | fveq2d 6775 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘𝐴) = (Base‘(𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉))) |
9 | 3, 8 | eqtr4id 2799 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 〈cop 4573 〈cotp 4575 × cxp 5588 ‘cfv 6432 (class class class)co 7271 Fincfn 8716 sSet csts 16862 ndxcnx 16892 Basecbs 16910 .rcmulr 16961 freeLMod cfrlm 20951 maMul cmmul 21530 Mat cmat 21552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-mulr 16974 df-mat 21553 |
This theorem is referenced by: mat0 21564 matinvg 21565 matbas2 21568 matplusg2 21574 matvsca2 21575 matlmod 21576 matsubg 21579 matsubgcell 21581 matgsum 21584 matdim 31694 matunitlindflem2 35770 matunitlindf 35771 |
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