Theorem grpodivf 31147

Description: Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdivf.1	⊢ 𝑋 = ran 𝐺
grpdivf.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
grpodivf	⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem grpodivf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpdivf.1	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
2		eqid 2760	. . . . . . . 8 ⊢ (inv‘𝐺) = (inv‘𝐺)
3	1, 2	grpoinvcl 31133	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
4	3	3adant2 1149	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
5	1	grpocl 31109	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
6	4, 5	syld3an3 1436	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
7	6	3expib 1140	. . . 4 ⊢ (𝐺 ∈ GrpOp → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋))
8	7	ralrimivv 3203	. . 3 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
9		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))
10	9	fmpo 8081	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
11	8, 10	sylib 221	. 2 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
12		grpdivf.3	. . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
13	1, 2, 12	grpodivfval 31143	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))))
14	13	feq1d 6694	. 2 ⊢ (𝐺 ∈ GrpOp → (𝐷:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋))
15	11, 14	mpbird 260	1 ⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076   × cxp 5648  ran crn 5651  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422   ∈ cmpo 7424  GrpOpcgr 31098  invcgn 31100   /_𝑔 cgs 31101

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-grpo 31102  df-gid 31103  df-ginv 31104  df-gdiv 31105

This theorem is used by:  grpodivcl  31148