Theorem grpodivf 28900

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 2738	. . . . . . . 8 ⊢ (inv‘𝐺) = (inv‘𝐺)
3	1, 2	grpoinvcl 28886	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
4	3	3adant2 1130	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
5	1	grpocl 28862	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
6	4, 5	syld3an3 1408	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
7	6	3expib 1121	. . . 4 ⊢ (𝐺 ∈ GrpOp → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋))
8	7	ralrimivv 3122	. . 3 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
9		eqid 2738	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))
10	9	fmpo 7908	. . 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 217	. 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 28896	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))))
14	13	feq1d 6585	. 2 ⊢ (𝐺 ∈ GrpOp → (𝐷:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋))
15	11, 14	mpbird 256	1 ⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064   × cxp 5587  ran crn 5590  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  GrpOpcgr 28851  invcgn 28853   /_𝑔 cgs 28854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ginv 28857  df-gdiv 28858

This theorem is referenced by:  grpodivcl  28901