Theorem ismndo2 33673

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 33670	. . . . 5 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))
3		rngopidOLD 33652	. . . . 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	syl5eq 2668	. . 3 ⊢ (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺)
6	5	a1i 11	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺))
7		fdm 6051	. . . . . 6 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
8	7	dmeqd 5326	. . . . 5 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom dom 𝐺 = dom (𝑋 × 𝑋))
9		dmxpid 5345	. . . . 5 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	syl6req 2673	. . . 4 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → 𝑋 = dom dom 𝐺)
11	10	3ad2ant1 1082	. . 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 2622	. . . 4 ⊢ dom dom 𝐺 = dom dom 𝐺
14	13	ismndo1 33672	. . 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 5347	. . . . . . 7 ⊢ ((𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺) ↔ 𝑋 = dom dom 𝐺)
16	15	biimpri 218	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺))
17		feq23 6029	. . . . . 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 703	. . . . 5 ⊢ (𝑋 = dom dom 𝐺 → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
19		raleq 3138	. . . . . . 7 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
20	19	raleqbi1dv 3146	. . . . . 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 3146	. . . . 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 3138	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ↔ ∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
23	22	rexeqbi1dv 3147	. . . . 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 1399	. . . 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 332	. . 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 237	. 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 369	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   × cxp 5112  dom cdm 5114  ran crn 5115  ⟶wf 5884  (class class class)co 6650   ExId cexid 33643  Magmacmagm 33647  MndOpcmndo 33665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666

This theorem is referenced by:  grpomndo  33674