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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgrpbase | Structured version Visualization version GIF version |
Description: The base set of the translation group is the set of all translations (for a fiducial co-atom π). (Contributed by NM, 5-Jun-2013.) |
Ref | Expression |
---|---|
tgrpset.h | β’ π» = (LHypβπΎ) |
tgrpset.t | β’ π = ((LTrnβπΎ)βπ) |
tgrpset.g | β’ πΊ = ((TGrpβπΎ)βπ) |
tgrp.c | β’ πΆ = (BaseβπΊ) |
Ref | Expression |
---|---|
tgrpbase | β’ ((πΎ β π β§ π β π») β πΆ = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgrpset.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | tgrpset.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
3 | tgrpset.g | . . . 4 β’ πΊ = ((TGrpβπΎ)βπ) | |
4 | 1, 2, 3 | tgrpset 40129 | . . 3 β’ ((πΎ β π β§ π β π») β πΊ = {β¨(Baseβndx), πβ©, β¨(+gβndx), (π β π, π β π β¦ (π β π))β©}) |
5 | 4 | fveq2d 6889 | . 2 β’ ((πΎ β π β§ π β π») β (BaseβπΊ) = (Baseβ{β¨(Baseβndx), πβ©, β¨(+gβndx), (π β π, π β π β¦ (π β π))β©})) |
6 | tgrp.c | . 2 β’ πΆ = (BaseβπΊ) | |
7 | 2 | fvexi 6899 | . . 3 β’ π β V |
8 | eqid 2726 | . . . 4 β’ {β¨(Baseβndx), πβ©, β¨(+gβndx), (π β π, π β π β¦ (π β π))β©} = {β¨(Baseβndx), πβ©, β¨(+gβndx), (π β π, π β π β¦ (π β π))β©} | |
9 | 8 | grpbase 17240 | . . 3 β’ (π β V β π = (Baseβ{β¨(Baseβndx), πβ©, β¨(+gβndx), (π β π, π β π β¦ (π β π))β©})) |
10 | 7, 9 | ax-mp 5 | . 2 β’ π = (Baseβ{β¨(Baseβndx), πβ©, β¨(+gβndx), (π β π, π β π β¦ (π β π))β©}) |
11 | 5, 6, 10 | 3eqtr4g 2791 | 1 β’ ((πΎ β π β§ π β π») β πΆ = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 {cpr 4625 β¨cop 4629 β ccom 5673 βcfv 6537 β cmpo 7407 ndxcnx 17135 Basecbs 17153 +gcplusg 17206 LHypclh 39368 LTrncltrn 39485 TGrpctgrp 40126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-tgrp 40127 |
This theorem is referenced by: tgrpgrplem 40133 tgrpabl 40135 |
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