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Theorem ablsub2inv 18436
Description: Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
Hypotheses
Ref Expression
ablsub2inv.b 𝐵 = (Base‘𝐺)
ablsub2inv.m = (-g𝐺)
ablsub2inv.n 𝑁 = (invg𝐺)
ablsub2inv.g (𝜑𝐺 ∈ Abel)
ablsub2inv.x (𝜑𝑋𝐵)
ablsub2inv.y (𝜑𝑌𝐵)
Assertion
Ref Expression
ablsub2inv (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))

Proof of Theorem ablsub2inv
StepHypRef Expression
1 ablsub2inv.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2760 . . 3 (+g𝐺) = (+g𝐺)
3 ablsub2inv.m . . 3 = (-g𝐺)
4 ablsub2inv.n . . 3 𝑁 = (invg𝐺)
5 ablsub2inv.g . . . 4 (𝜑𝐺 ∈ Abel)
6 ablgrp 18418 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
75, 6syl 17 . . 3 (𝜑𝐺 ∈ Grp)
8 ablsub2inv.x . . . 4 (𝜑𝑋𝐵)
91, 4grpinvcl 17688 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
107, 8, 9syl2anc 696 . . 3 (𝜑 → (𝑁𝑋) ∈ 𝐵)
11 ablsub2inv.y . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 7, 10, 11grpsubinv 17709 . 2 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = ((𝑁𝑋)(+g𝐺)𝑌))
131, 2ablcom 18430 . . . . . 6 ((𝐺 ∈ Abel ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵) → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
145, 10, 11, 13syl3anc 1477 . . . . 5 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
151, 4grpinvinv 17703 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
167, 11, 15syl2anc 696 . . . . . 6 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1716oveq1d 6829 . . . . 5 (𝜑 → ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)) = (𝑌(+g𝐺)(𝑁𝑋)))
1814, 17eqtr4d 2797 . . . 4 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
191, 4grpinvcl 17688 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
207, 11, 19syl2anc 696 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
211, 2, 4grpinvadd 17714 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
227, 8, 20, 21syl3anc 1477 . . . 4 (𝜑 → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
2318, 22eqtr4d 2797 . . 3 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
241, 2, 4, 3grpsubval 17686 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
258, 11, 24syl2anc 696 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
2625fveq2d 6357 . . 3 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
2723, 26eqtr4d 2797 . 2 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋 𝑌)))
281, 3, 4grpinvsub 17718 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
297, 8, 11, 28syl3anc 1477 . 2 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
3012, 27, 293eqtrd 2798 1 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  Grpcgrp 17643  invgcminusg 17644  -gcsg 17645  Abelcabl 18414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-sbg 17648  df-cmn 18415  df-abl 18416
This theorem is referenced by:  ngpinvds  22638  hdmap1neglem1N  37637
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