Theorem ismndo2 35675

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 35672	. . . . 5 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))
3		rngopidOLD 35654	. . . . 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 2785	. . 3 ⊢ (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺)
6	5	a1i 11	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺))
7		fdm 6513	. . . . . 6 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
8	7	dmeqd 5748	. . . . 5 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom dom 𝐺 = dom (𝑋 × 𝑋))
9		dmxpid 5773	. . . . 5 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	eqtr2di 2790	. . . 4 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → 𝑋 = dom dom 𝐺)
11	10	3ad2ant1 1134	. . 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 35674	. . 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 5775	. . . . . . 7 ⊢ ((𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺) ↔ 𝑋 = dom dom 𝐺)
16	15	biimpri 231	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺))
17		feq23 6488	. . . . . 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 3310	. . . . . . 7 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
20	19	raleqbi1dv 3308	. . . . . 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 3308	. . . . 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 3310	. . . . . 6 ⊢ (𝑋 = dom dom 𝐺 → (∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ↔ ∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
23	22	rexeqbi1dv 3309	. . . . 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 1437	. . . 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 346	. . 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 250	. 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 384	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 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3053  ∃wrex 3054   ∩ cin 3842   × cxp 5523  dom cdm 5525  ran crn 5526  ⟶wf 6335  (class class class)co 7170   ExId cexid 35645  Magmacmagm 35649  MndOpcmndo 35667

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347  df-ov 7173  df-ass 35644  df-exid 35646  df-mgmOLD 35650  df-sgrOLD 35662  df-mndo 35668

This theorem is referenced by:  grpomndo  35676