Theorem grpodivf 30627

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 2739	. . . . . . . 8 ⊢ (inv‘𝐺) = (inv‘𝐺)
3	1, 2	grpoinvcl 30613	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
4	3	3adant2 1137	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
5	1	grpocl 30589	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
6	4, 5	syld3an3 1417	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
7	6	3expib 1128	. . . 4 ⊢ (𝐺 ∈ GrpOp → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋))
8	7	ralrimivv 3180	. . 3 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
9		eqid 2739	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))
10	9	fmpo 8010	. . 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 219	. 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 30623	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))))
14	13	feq1d 6637	. 2 ⊢ (𝐺 ∈ GrpOp → (𝐷:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋))
15	11, 14	mpbird 258	1 ⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3053   × cxp 5616  ran crn 5619  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  GrpOpcgr 30578  invcgn 30580   /_𝑔 cgs 30581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-grpo 30582  df-gid 30583  df-ginv 30584  df-gdiv 30585

This theorem is referenced by:  grpodivcl  30628