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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 | ⊢ 𝑋 = (Base‘𝐺) |
ngpinvds.i | ⊢ 𝐼 = (invg‘𝐺) |
ngpinvds.d | ⊢ 𝐷 = (dist‘𝐺) |
Ref | Expression |
---|---|
ngpinvds | ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐼‘𝐴)𝐷(𝐼‘𝐵)) = (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpinvds.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
2 | eqid 2725 | . . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | ngpinvds.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
4 | simplr 767 | . . . 4 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐺 ∈ Abel) | |
5 | simprr 771 | . . . 4 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
6 | simprl 769 | . . . 4 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
7 | 1, 2, 3, 4, 5, 6 | ablsub2inv 19775 | . . 3 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐼‘𝐵)(-g‘𝐺)(𝐼‘𝐴)) = (𝐴(-g‘𝐺)𝐵)) |
8 | 7 | fveq2d 6900 | . 2 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((norm‘𝐺)‘((𝐼‘𝐵)(-g‘𝐺)(𝐼‘𝐴))) = ((norm‘𝐺)‘(𝐴(-g‘𝐺)𝐵))) |
9 | simpll 765 | . . 3 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐺 ∈ NrmGrp) | |
10 | ngpgrp 24552 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐺 ∈ Grp) |
12 | 1, 3 | grpinvcl 18952 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐼‘𝐴) ∈ 𝑋) |
13 | 11, 6, 12 | syl2anc 582 | . . 3 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐼‘𝐴) ∈ 𝑋) |
14 | 1, 3 | grpinvcl 18952 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋) → (𝐼‘𝐵) ∈ 𝑋) |
15 | 11, 5, 14 | syl2anc 582 | . . 3 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐼‘𝐵) ∈ 𝑋) |
16 | eqid 2725 | . . . 4 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
17 | ngpinvds.d | . . . 4 ⊢ 𝐷 = (dist‘𝐺) | |
18 | 16, 1, 2, 17 | ngpdsr 24558 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐼‘𝐴) ∈ 𝑋 ∧ (𝐼‘𝐵) ∈ 𝑋) → ((𝐼‘𝐴)𝐷(𝐼‘𝐵)) = ((norm‘𝐺)‘((𝐼‘𝐵)(-g‘𝐺)(𝐼‘𝐴)))) |
19 | 9, 13, 15, 18 | syl3anc 1368 | . 2 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐼‘𝐴)𝐷(𝐼‘𝐵)) = ((norm‘𝐺)‘((𝐼‘𝐵)(-g‘𝐺)(𝐼‘𝐴)))) |
20 | 16, 1, 2, 17 | ngpds 24557 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((norm‘𝐺)‘(𝐴(-g‘𝐺)𝐵))) |
21 | 9, 6, 5, 20 | syl3anc 1368 | . 2 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = ((norm‘𝐺)‘(𝐴(-g‘𝐺)𝐵))) |
22 | 8, 19, 21 | 3eqtr4d 2775 | 1 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐼‘𝐴)𝐷(𝐼‘𝐵)) = (𝐴𝐷𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 distcds 17245 Grpcgrp 18898 invgcminusg 18899 -gcsg 18900 Abelcabl 19748 normcnm 24529 NrmGrpcngp 24530 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-0g 17426 df-topgen 17428 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-cmn 19749 df-abl 19750 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-xms 24270 df-ms 24271 df-nm 24535 df-ngp 24536 |
This theorem is referenced by: ngptgp 24589 |
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