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Mirrors > Home > MPE Home > Th. List > mgpds | Structured version Visualization version GIF version |
Description: Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
mgpds.2 | ⊢ 𝐵 = (dist‘𝑅) |
Ref | Expression |
---|---|
mgpds | ⊢ 𝐵 = (dist‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpds.2 | . 2 ⊢ 𝐵 = (dist‘𝑅) | |
2 | mgpbas.1 | . . . 4 ⊢ 𝑀 = (mulGrp‘𝑅) | |
3 | eqid 2728 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 2, 3 | mgpval 20076 | . . 3 ⊢ 𝑀 = (𝑅 sSet ⟨(+g‘ndx), (.r‘𝑅)⟩) |
5 | dsid 17366 | . . 3 ⊢ dist = Slot (dist‘ndx) | |
6 | dsndxnplusgndx 17370 | . . 3 ⊢ (dist‘ndx) ≠ (+g‘ndx) | |
7 | 4, 5, 6 | setsplusg 19300 | . 2 ⊢ (dist‘𝑅) = (dist‘𝑀) |
8 | 1, 7 | eqtri 2756 | 1 ⊢ 𝐵 = (dist‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ‘cfv 6548 .rcmulr 17233 distcds 17241 mulGrpcmgp 20073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-sets 17132 df-slot 17150 df-ndx 17162 df-plusg 17245 df-ds 17254 df-mgp 20074 |
This theorem is referenced by: (None) |
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