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Theorem ablsub2inv 19806
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 2726 . . 3 (+g𝐺) = (+g𝐺)
3 ablsub2inv.m . . 3 = (-g𝐺)
4 ablsub2inv.n . . 3 𝑁 = (invg𝐺)
5 ablsub2inv.g . . . 4 (𝜑𝐺 ∈ Abel)
6 ablgrp 19783 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
75, 6syl 17 . . 3 (𝜑𝐺 ∈ Grp)
8 ablsub2inv.x . . . 4 (𝜑𝑋𝐵)
91, 4grpinvcl 18982 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
107, 8, 9syl2anc 582 . . 3 (𝜑 → (𝑁𝑋) ∈ 𝐵)
11 ablsub2inv.y . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 7, 10, 11grpsubinv 19006 . 2 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = ((𝑁𝑋)(+g𝐺)𝑌))
131, 2ablcom 19797 . . . . . 6 ((𝐺 ∈ Abel ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵) → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
145, 10, 11, 13syl3anc 1368 . . . . 5 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
151, 4grpinvinv 19000 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
167, 11, 15syl2anc 582 . . . . . 6 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1716oveq1d 7439 . . . . 5 (𝜑 → ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)) = (𝑌(+g𝐺)(𝑁𝑋)))
1814, 17eqtr4d 2769 . . . 4 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
191, 4grpinvcl 18982 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
207, 11, 19syl2anc 582 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
211, 2, 4grpinvadd 19012 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
227, 8, 20, 21syl3anc 1368 . . . 4 (𝜑 → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
2318, 22eqtr4d 2769 . . 3 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
241, 2, 4, 3grpsubval 18980 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
258, 11, 24syl2anc 582 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
2625fveq2d 6905 . . 3 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
2723, 26eqtr4d 2769 . 2 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋 𝑌)))
281, 3, 4grpinvsub 19016 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
297, 8, 11, 28syl3anc 1368 . 2 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
3012, 27, 293eqtrd 2770 1 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6554  (class class class)co 7424  Basecbs 17213  +gcplusg 17266  Grpcgrp 18928  invgcminusg 18929  -gcsg 18930  Abelcabl 19779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  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-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-0g 17456  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-sbg 18933  df-cmn 19780  df-abl 19781
This theorem is referenced by:  ngpinvds  24613
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