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Theorem ablsub2inv 18423
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 2771 . . 3 (+g𝐺) = (+g𝐺)
3 ablsub2inv.m . . 3 = (-g𝐺)
4 ablsub2inv.n . . 3 𝑁 = (invg𝐺)
5 ablsub2inv.g . . . 4 (𝜑𝐺 ∈ Abel)
6 ablgrp 18405 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
75, 6syl 17 . . 3 (𝜑𝐺 ∈ Grp)
8 ablsub2inv.x . . . 4 (𝜑𝑋𝐵)
91, 4grpinvcl 17675 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
107, 8, 9syl2anc 573 . . 3 (𝜑 → (𝑁𝑋) ∈ 𝐵)
11 ablsub2inv.y . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 7, 10, 11grpsubinv 17696 . 2 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = ((𝑁𝑋)(+g𝐺)𝑌))
131, 2ablcom 18417 . . . . . 6 ((𝐺 ∈ Abel ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵) → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
145, 10, 11, 13syl3anc 1476 . . . . 5 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
151, 4grpinvinv 17690 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
167, 11, 15syl2anc 573 . . . . . 6 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1716oveq1d 6811 . . . . 5 (𝜑 → ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)) = (𝑌(+g𝐺)(𝑁𝑋)))
1814, 17eqtr4d 2808 . . . 4 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
191, 4grpinvcl 17675 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
207, 11, 19syl2anc 573 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
211, 2, 4grpinvadd 17701 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
227, 8, 20, 21syl3anc 1476 . . . 4 (𝜑 → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
2318, 22eqtr4d 2808 . . 3 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
241, 2, 4, 3grpsubval 17673 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
258, 11, 24syl2anc 573 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
2625fveq2d 6337 . . 3 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
2723, 26eqtr4d 2808 . 2 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋 𝑌)))
281, 3, 4grpinvsub 17705 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
297, 8, 11, 28syl3anc 1476 . 2 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
3012, 27, 293eqtrd 2809 1 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  Grpcgrp 17630  invgcminusg 17631  -gcsg 17632  Abelcabl 18401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-cmn 18402  df-abl 18403
This theorem is referenced by:  ngpinvds  22637
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