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Mirrors > Home > MPE Home > Th. List > matplusg | Structured version Visualization version GIF version |
Description: The matrix ring has the same addition 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 |
---|---|
matplusg | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matbas.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | matbas.g | . . . 4 ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) | |
3 | eqid 2799 | . . . 4 ⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
4 | 1, 2, 3 | matval 20542 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉)) |
5 | 4 | fveq2d 6415 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (+g‘𝐴) = (+g‘(𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉))) |
6 | plusgid 16298 | . . 3 ⊢ +g = Slot (+g‘ndx) | |
7 | plusgndx 16297 | . . . 4 ⊢ (+g‘ndx) = 2 | |
8 | 2re 11387 | . . . . . 6 ⊢ 2 ∈ ℝ | |
9 | 2lt3 11492 | . . . . . 6 ⊢ 2 < 3 | |
10 | 8, 9 | ltneii 10440 | . . . . 5 ⊢ 2 ≠ 3 |
11 | mulrndx 16317 | . . . . 5 ⊢ (.r‘ndx) = 3 | |
12 | 10, 11 | neeqtrri 3044 | . . . 4 ⊢ 2 ≠ (.r‘ndx) |
13 | 7, 12 | eqnetri 3041 | . . 3 ⊢ (+g‘ndx) ≠ (.r‘ndx) |
14 | 6, 13 | setsnid 16240 | . 2 ⊢ (+g‘𝐺) = (+g‘(𝐺 sSet 〈(.r‘ndx), (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)〉)) |
15 | 5, 14 | syl6reqr 2852 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 〈cop 4374 〈cotp 4376 × cxp 5310 ‘cfv 6101 (class class class)co 6878 Fincfn 8195 2c2 11368 3c3 11369 ndxcnx 16181 sSet csts 16182 +gcplusg 16267 .rcmulr 16268 freeLMod cfrlm 20415 maMul cmmul 20514 Mat cmat 20538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-ot 4377 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-ndx 16187 df-slot 16188 df-sets 16191 df-plusg 16280 df-mulr 16281 df-mat 20539 |
This theorem is referenced by: mat0 20548 matinvg 20549 matplusg2 20558 matlmod 20560 matsubg 20563 matgsum 20568 |
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