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Theorem grpsubfval 17586
Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
grpsubval.b 𝐵 = (Base‘𝐺)
grpsubval.p + = (+g𝐺)
grpsubval.i 𝐼 = (invg𝐺)
grpsubval.m = (-g𝐺)
Assertion
Ref Expression
grpsubfval = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐼,𝑦   𝑥, + ,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem grpsubfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpsubval.m . . 3 = (-g𝐺)
2 fveq2 6304 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpsubval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3syl6eqr 2776 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6304 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpsubval.p . . . . . . 7 + = (+g𝐺)
75, 6syl6eqr 2776 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
8 eqidd 2725 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
9 fveq2 6304 . . . . . . . 8 (𝑔 = 𝐺 → (invg𝑔) = (invg𝐺))
10 grpsubval.i . . . . . . . 8 𝐼 = (invg𝐺)
119, 10syl6eqr 2776 . . . . . . 7 (𝑔 = 𝐺 → (invg𝑔) = 𝐼)
1211fveq1d 6306 . . . . . 6 (𝑔 = 𝐺 → ((invg𝑔)‘𝑦) = (𝐼𝑦))
137, 8, 12oveq123d 6786 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)((invg𝑔)‘𝑦)) = (𝑥 + (𝐼𝑦)))
144, 4, 13mpt2eq123dv 6834 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
15 df-sbg 17549 . . . 4 -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
16 fvex 6314 . . . . . 6 (Base‘𝐺) ∈ V
173, 16eqeltri 2799 . . . . 5 𝐵 ∈ V
1817, 17mpt2ex 7367 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V
1914, 15, 18fvmpt 6396 . . 3 (𝐺 ∈ V → (-g𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
201, 19syl5eq 2770 . 2 (𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
21 fvprc 6298 . . . 4 𝐺 ∈ V → (-g𝐺) = ∅)
221, 21syl5eq 2770 . . 3 𝐺 ∈ V → = ∅)
23 fvprc 6298 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23syl5eq 2770 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
25 mpt2eq12 6832 . . . . 5 ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼𝑦))))
2624, 24, 25syl2anc 696 . . . 4 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼𝑦))))
27 mpt20 6842 . . . 4 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼𝑦))) = ∅
2826, 27syl6eq 2774 . . 3 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
2922, 28eqtr4d 2761 . 2 𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
3020, 29pm2.61i 176 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1596  wcel 2103  Vcvv 3304  c0 4023  cfv 6001  (class class class)co 6765  cmpt2 6767  Basecbs 15980  +gcplusg 16064  invgcminusg 17545  -gcsg 17546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-sbg 17549
This theorem is referenced by:  grpsubval  17587  grpsubf  17616  grpsubpropd  17642  grpsubpropd2  17643  tgpsubcn  22016  tngtopn  22576
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