Theorem ismndo2 36032

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 36029	. . . . 5 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))
3		rngopidOLD 36011	. . . . 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 2790	. . 3 ⊢ (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺)
6	5	a1i 11	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺))
7		fdm 6609	. . . . . 6 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
8	7	dmeqd 5814	. . . . 5 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom dom 𝐺 = dom (𝑋 × 𝑋))
9		dmxpid 5839	. . . . 5 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	eqtr2di 2795	. . . 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 2738	. . . 4 ⊢ dom dom 𝐺 = dom dom 𝐺
14	13	ismndo1 36031	. . 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 5841	. . . . . . 7 ⊢ ((𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺) ↔ 𝑋 = dom dom 𝐺)
16	15	biimpri 227	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺))
17		feq23 6584	. . . . . 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 685	. . . . 5 ⊢ (𝑋 = dom dom 𝐺 → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
19		raleq 3342	. . . . . . 7 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
20	19	raleqbi1dv 3340	. . . . . 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 3340	. . . . 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 3342	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ↔ ∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
23	22	rexeqbi1dv 3341	. . . . 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 343	. . 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 246	. 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 381	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   × cxp 5587  dom cdm 5589  ran crn 5590  ⟶wf 6429  (class class class)co 7275   ExId cexid 36002  Magmacmagm 36006  MndOpcmndo 36024

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-sep 5223  ax-nul 5230  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-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-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-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-ass 36001  df-exid 36003  df-mgmOLD 36007  df-sgrOLD 36019  df-mndo 36025

This theorem is referenced by:  grpomndo  36033