Theorem ismndo2 37861

Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ismndo2.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
ismndo2	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))

Distinct variable groups:   𝑥,𝐺,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem ismndo2

Step	Hyp	Ref	Expression
1		ismndo2.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2		mndomgmid 37858	. . . . 5 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))
3		rngopidOLD 37840	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
4	2, 3	syl 17	. . . 4 ⊢ (𝐺 ∈ MndOp → ran 𝐺 = dom dom 𝐺)
5	1, 4	eqtrid 2787	. . 3 ⊢ (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺)
6	5	a1i 11	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺))
7		fdm 6746	. . . . . 6 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
8	7	dmeqd 5919	. . . . 5 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom dom 𝐺 = dom (𝑋 × 𝑋))
9		dmxpid 5944	. . . . 5 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	eqtr2di 2792	. . . 4 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → 𝑋 = dom dom 𝐺)
11	10	3ad2ant1 1132	. . 3 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) → 𝑋 = dom dom 𝐺)
12	11	a1i 11	. 2 ⊢ (𝐺 ∈ 𝐴 → ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) → 𝑋 = dom dom 𝐺))
13		eqid 2735	. . . 4 ⊢ dom dom 𝐺 = dom dom 𝐺
14	13	ismndo1 37860	. . 3 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ∧ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
15		xpid11 5946	. . . . . . 7 ⊢ ((𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺) ↔ 𝑋 = dom dom 𝐺)
16	15	biimpri 228	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺))
17		feq23 6720	. . . . . 6 ⊢ (((𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺) ∧ 𝑋 = dom dom 𝐺) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
18	16, 17	mpancom 688	. . . . 5 ⊢ (𝑋 = dom dom 𝐺 → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
19		raleq 3321	. . . . . . 7 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
20	19	raleqbi1dv 3336	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
21	20	raleqbi1dv 3336	. . . . 5 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
22		raleq 3321	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ↔ ∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
23	22	rexeqbi1dv 3337	. . . . 5 ⊢ (𝑋 = dom dom 𝐺 → (∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ↔ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
24	18, 21, 23	3anbi123d 1435	. . . 4 ⊢ (𝑋 = dom dom 𝐺 → ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) ↔ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ∧ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
25	24	bibi2d 342	. . 3 ⊢ (𝑋 = dom dom 𝐺 → ((𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) ↔ (𝐺 ∈ MndOp ↔ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ∧ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))))
26	14, 25	syl5ibrcom 247	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝑋 = dom dom 𝐺 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))))
27	6, 12, 26	pm5.21ndd 379	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∩ cin 3962   × cxp 5687  dom cdm 5689  ran crn 5690  ⟶wf 6559  (class class class)co 7431   ExId cexid 37831  Magmacmagm 37835  MndOpcmndo 37853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-ass 37830  df-exid 37832  df-mgmOLD 37836  df-sgrOLD 37848  df-mndo 37854

This theorem is referenced by:  grpomndo  37862