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Theorem ablsub2inv 19826
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 2737 . . 3 (+g𝐺) = (+g𝐺)
3 ablsub2inv.m . . 3 = (-g𝐺)
4 ablsub2inv.n . . 3 𝑁 = (invg𝐺)
5 ablsub2inv.g . . . 4 (𝜑𝐺 ∈ Abel)
6 ablgrp 19803 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
75, 6syl 17 . . 3 (𝜑𝐺 ∈ Grp)
8 ablsub2inv.x . . . 4 (𝜑𝑋𝐵)
91, 4grpinvcl 19005 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
107, 8, 9syl2anc 584 . . 3 (𝜑 → (𝑁𝑋) ∈ 𝐵)
11 ablsub2inv.y . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 7, 10, 11grpsubinv 19030 . 2 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = ((𝑁𝑋)(+g𝐺)𝑌))
131, 2ablcom 19817 . . . . . 6 ((𝐺 ∈ Abel ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵) → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
145, 10, 11, 13syl3anc 1373 . . . . 5 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
151, 4grpinvinv 19023 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
167, 11, 15syl2anc 584 . . . . . 6 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1716oveq1d 7446 . . . . 5 (𝜑 → ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)) = (𝑌(+g𝐺)(𝑁𝑋)))
1814, 17eqtr4d 2780 . . . 4 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
191, 4grpinvcl 19005 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
207, 11, 19syl2anc 584 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
211, 2, 4grpinvadd 19036 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
227, 8, 20, 21syl3anc 1373 . . . 4 (𝜑 → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
2318, 22eqtr4d 2780 . . 3 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
241, 2, 4, 3grpsubval 19003 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
258, 11, 24syl2anc 584 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
2625fveq2d 6910 . . 3 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
2723, 26eqtr4d 2780 . 2 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋 𝑌)))
281, 3, 4grpinvsub 19040 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
297, 8, 11, 28syl3anc 1373 . 2 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
3012, 27, 293eqtrd 2781 1 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  invgcminusg 18952  -gcsg 18953  Abelcabl 19799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-cmn 19800  df-abl 19801
This theorem is referenced by:  ngpinvds  24626
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