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Mirrors > Home > ILE Home > Th. List > mgpdsg | GIF version |
Description: Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | β’ π = (mulGrpβπ ) |
mgpds.2 | β’ π΅ = (distβπ ) |
Ref | Expression |
---|---|
mgpdsg | β’ (π β π β π΅ = (distβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpds.2 | . 2 β’ π΅ = (distβπ ) | |
2 | mulrslid 12589 | . . . . 5 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
3 | 2 | slotex 12488 | . . . 4 β’ (π β π β (.rβπ ) β V) |
4 | dsslid 12667 | . . . . 5 β’ (dist = Slot (distβndx) β§ (distβndx) β β) | |
5 | dsndxnplusgndx 12671 | . . . . 5 β’ (distβndx) β (+gβndx) | |
6 | plusgslid 12570 | . . . . . 6 β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | |
7 | 6 | simpri 113 | . . . . 5 β’ (+gβndx) β β |
8 | 4, 5, 7 | setsslnid 12513 | . . . 4 β’ ((π β π β§ (.rβπ ) β V) β (distβπ ) = (distβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
9 | 3, 8 | mpdan 421 | . . 3 β’ (π β π β (distβπ ) = (distβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
10 | mgpbas.1 | . . . . 5 β’ π = (mulGrpβπ ) | |
11 | eqid 2177 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
12 | 10, 11 | mgpvalg 13131 | . . . 4 β’ (π β π β π = (π sSet β¨(+gβndx), (.rβπ )β©)) |
13 | 12 | fveq2d 5519 | . . 3 β’ (π β π β (distβπ) = (distβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
14 | 9, 13 | eqtr4d 2213 | . 2 β’ (π β π β (distβπ ) = (distβπ)) |
15 | 1, 14 | eqtrid 2222 | 1 β’ (π β π β π΅ = (distβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2737 β¨cop 3595 βcfv 5216 (class class class)co 5874 βcn 8918 ndxcnx 12458 sSet csts 12459 Slot cslot 12460 +gcplusg 12535 .rcmulr 12536 distcds 12544 mulGrpcmgp 13128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-5 8980 df-6 8981 df-7 8982 df-8 8983 df-9 8984 df-n0 9176 df-z 9253 df-dec 9384 df-ndx 12464 df-slot 12465 df-sets 12468 df-plusg 12548 df-mulr 12549 df-ds 12557 df-mgp 13129 |
This theorem is referenced by: (None) |
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