Theorem grpsubfval 17464

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	⊢ + = (+g‘𝐺)
grpsubval.i	⊢ 𝐼 = (invg‘𝐺)
grpsubval.m	⊢ − = (-g‘𝐺)

Assertion

Ref	Expression
grpsubfval	⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))

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.

Step	Hyp	Ref	Expression
1		grpsubval.m	. . 3 ⊢ − = (-g‘𝐺)
2		fveq2 6191	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpsubval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	syl6eqr 2674	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6191	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpsubval.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	syl6eqr 2674	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8		eqidd 2623	. . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥)
9		fveq2 6191	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺))
10		grpsubval.i	. . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺)
11	9, 10	syl6eqr 2674	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼)
12	11	fveq1d 6193	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦))
13	7, 8, 12	oveq123d 6671	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦)))
14	4, 4, 13	mpt2eq123dv 6717	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
15		df-sbg 17427	. . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
16		fvex 6201	. . . . . 6 ⊢ (Base‘𝐺) ∈ V
17	3, 16	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
18	17, 17	mpt2ex 7247	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V
19	14, 15, 18	fvmpt 6282	. . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
20	1, 19	syl5eq 2668	. 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
21		fvprc 6185	. . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅)
22	1, 21	syl5eq 2668	. . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅)
23		fvprc 6185	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
24	3, 23	syl5eq 2668	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
25		mpt2eq12 6715	. . . . 5 ⊢ ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼‘𝑦))))
26	24, 24, 25	syl2anc 693	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼‘𝑦))))
27		mpt20 6725	. . . 4 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼‘𝑦))) = ∅
28	26, 27	syl6eq 2672	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = ∅)
29	22, 28	eqtr4d 2659	. 2 ⊢ (¬ 𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
30	20, 29	pm2.61i 176	1 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  +_gcplusg 15941  inv_gcminusg 17423  -_gcsg 17424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-sbg 17427

This theorem is referenced by:  grpsubval  17465  grpsubf  17494  grpsubpropd  17520  grpsubpropd2  17521  tgpsubcn  21894  tngtopn  22454