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| Mirrors > Home > MPE Home > Th. List > ngpinvds | Structured version Visualization version GIF version | ||
| Description: Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| Ref | Expression |
|---|---|
| ngpinvds.x | β’ π = (BaseβπΊ) |
| ngpinvds.i | β’ πΌ = (invgβπΊ) |
| ngpinvds.d | β’ π· = (distβπΊ) |
| Ref | Expression |
|---|---|
| ngpinvds | β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β ((πΌβπ΄)π·(πΌβπ΅)) = (π΄π·π΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ngpinvds.x | . . . 4 β’ π = (BaseβπΊ) | |
| 2 | eqid 2728 | . . . 4 β’ (-gβπΊ) = (-gβπΊ) | |
| 3 | ngpinvds.i | . . . 4 β’ πΌ = (invgβπΊ) | |
| 4 | simplr 767 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β πΊ β Abel) | |
| 5 | simprr 771 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β π΅ β π) | |
| 6 | simprl 769 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β π΄ β π) | |
| 7 | 1, 2, 3, 4, 5, 6 | ablsub2inv 19770 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β ((πΌβπ΅)(-gβπΊ)(πΌβπ΄)) = (π΄(-gβπΊ)π΅)) |
| 8 | 7 | fveq2d 6906 | . 2 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β ((normβπΊ)β((πΌβπ΅)(-gβπΊ)(πΌβπ΄))) = ((normβπΊ)β(π΄(-gβπΊ)π΅))) |
| 9 | simpll 765 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β πΊ β NrmGrp) | |
| 10 | ngpgrp 24528 | . . . . 5 β’ (πΊ β NrmGrp β πΊ β Grp) | |
| 11 | 9, 10 | syl 17 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β πΊ β Grp) |
| 12 | 1, 3 | grpinvcl 18951 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π) β (πΌβπ΄) β π) |
| 13 | 11, 6, 12 | syl2anc 582 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β (πΌβπ΄) β π) |
| 14 | 1, 3 | grpinvcl 18951 | . . . 4 β’ ((πΊ β Grp β§ π΅ β π) β (πΌβπ΅) β π) |
| 15 | 11, 5, 14 | syl2anc 582 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β (πΌβπ΅) β π) |
| 16 | eqid 2728 | . . . 4 β’ (normβπΊ) = (normβπΊ) | |
| 17 | ngpinvds.d | . . . 4 β’ π· = (distβπΊ) | |
| 18 | 16, 1, 2, 17 | ngpdsr 24534 | . . 3 β’ ((πΊ β NrmGrp β§ (πΌβπ΄) β π β§ (πΌβπ΅) β π) β ((πΌβπ΄)π·(πΌβπ΅)) = ((normβπΊ)β((πΌβπ΅)(-gβπΊ)(πΌβπ΄)))) |
| 19 | 9, 13, 15, 18 | syl3anc 1368 | . 2 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β ((πΌβπ΄)π·(πΌβπ΅)) = ((normβπΊ)β((πΌβπ΅)(-gβπΊ)(πΌβπ΄)))) |
| 20 | 16, 1, 2, 17 | ngpds 24533 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = ((normβπΊ)β(π΄(-gβπΊ)π΅))) |
| 21 | 9, 6, 5, 20 | syl3anc 1368 | . 2 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) = ((normβπΊ)β(π΄(-gβπΊ)π΅))) |
| 22 | 8, 19, 21 | 3eqtr4d 2778 | 1 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β ((πΌβπ΄)π·(πΌβπ΅)) = (π΄π·π΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 Basecbs 17187 distcds 17249 Grpcgrp 18897 invgcminusg 18898 -gcsg 18899 Abelcabl 19743 normcnm 24505 NrmGrpcngp 24506 |
| 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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-0g 17430 df-topgen 17432 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-cmn 19744 df-abl 19745 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-xms 24246 df-ms 24247 df-nm 24511 df-ngp 24512 |
| This theorem is referenced by: ngptgp 24565 |
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