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Theorem ablsub2inv 18606
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 2778 . . 3 (+g𝐺) = (+g𝐺)
3 ablsub2inv.m . . 3 = (-g𝐺)
4 ablsub2inv.n . . 3 𝑁 = (invg𝐺)
5 ablsub2inv.g . . . 4 (𝜑𝐺 ∈ Abel)
6 ablgrp 18588 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
75, 6syl 17 . . 3 (𝜑𝐺 ∈ Grp)
8 ablsub2inv.x . . . 4 (𝜑𝑋𝐵)
91, 4grpinvcl 17858 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
107, 8, 9syl2anc 579 . . 3 (𝜑 → (𝑁𝑋) ∈ 𝐵)
11 ablsub2inv.y . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 7, 10, 11grpsubinv 17879 . 2 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = ((𝑁𝑋)(+g𝐺)𝑌))
131, 2ablcom 18600 . . . . . 6 ((𝐺 ∈ Abel ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵) → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
145, 10, 11, 13syl3anc 1439 . . . . 5 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑌(+g𝐺)(𝑁𝑋)))
151, 4grpinvinv 17873 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
167, 11, 15syl2anc 579 . . . . . 6 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1716oveq1d 6939 . . . . 5 (𝜑 → ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)) = (𝑌(+g𝐺)(𝑁𝑋)))
1814, 17eqtr4d 2817 . . . 4 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
191, 4grpinvcl 17858 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
207, 11, 19syl2anc 579 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
211, 2, 4grpinvadd 17884 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
227, 8, 20, 21syl3anc 1439 . . . 4 (𝜑 → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
2318, 22eqtr4d 2817 . . 3 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
241, 2, 4, 3grpsubval 17856 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
258, 11, 24syl2anc 579 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
2625fveq2d 6452 . . 3 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
2723, 26eqtr4d 2817 . 2 (𝜑 → ((𝑁𝑋)(+g𝐺)𝑌) = (𝑁‘(𝑋 𝑌)))
281, 3, 4grpinvsub 17888 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
297, 8, 11, 28syl3anc 1439 . 2 (𝜑 → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
3012, 27, 293eqtrd 2818 1 (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cfv 6137  (class class class)co 6924  Basecbs 16259  +gcplusg 16342  Grpcgrp 17813  invgcminusg 17814  -gcsg 17815  Abelcabl 18584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-0g 16492  df-mgm 17632  df-sgrp 17674  df-mnd 17685  df-grp 17816  df-minusg 17817  df-sbg 17818  df-cmn 18585  df-abl 18586
This theorem is referenced by:  ngpinvds  22829
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