Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 2726 | . . . 4 β’ (-gβπΊ) = (-gβπΊ) | |
| 3 | ngpinvds.i | . . . 4 β’ πΌ = (invgβπΊ) | |
| 4 | simplr 766 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β πΊ β Abel) | |
| 5 | simprr 770 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β π΅ β π) | |
| 6 | simprl 768 | . . . 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 19728 | . . 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 6889 | . 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 764 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β πΊ β NrmGrp) | |
| 10 | ngpgrp 24463 | . . . . 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 18917 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π) β (πΌβπ΄) β π) |
| 13 | 11, 6, 12 | syl2anc 583 | . . 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 18917 | . . . 4 β’ ((πΊ β Grp β§ π΅ β π) β (πΌβπ΅) β π) |
| 15 | 11, 5, 14 | syl2anc 583 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π)) β (πΌβπ΅) β π) |
| 16 | eqid 2726 | . . . 4 β’ (normβπΊ) = (normβπΊ) | |
| 17 | ngpinvds.d | . . . 4 β’ π· = (distβπΊ) | |
| 18 | 16, 1, 2, 17 | ngpdsr 24469 | . . 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 24468 | . . 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 2776 | 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 395 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 Basecbs 17153 distcds 17215 Grpcgrp 18863 invgcminusg 18864 -gcsg 18865 Abelcabl 19701 normcnm 24440 NrmGrpcngp 24441 |
| 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-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 ax-pre-sup 11190 |
| 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-rmo 3370 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-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-0g 17396 df-topgen 17398 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-cmn 19702 df-abl 19703 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-xms 24181 df-ms 24182 df-nm 24446 df-ngp 24447 |
| This theorem is referenced by: ngptgp 24500 |
| Copyright terms: Public domain | W3C validator |