Theorem grpomndo 35313

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

Assertion

Ref	Expression
grpomndo	⊢ (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp)

Proof of Theorem grpomndo

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2798	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
2	1	isgrpo 28280	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑤 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑤𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑤))))
3	2	biimpd 232	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑤 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑤𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑤))))
4	1	grpoidinv 28291	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)))
5		simpl 486	. . . . . . . . . . 11 ⊢ ((((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)) → ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))
6	5	ralimi 3128	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ran 𝐺(((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)) → ∀𝑦 ∈ ran 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))
7	6	reximi 3206	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)) → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))
8	1	ismndo2 35312	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ GrpOp → (𝐺 ∈ MndOp ↔ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
9	8	biimprcd 253	. . . . . . . . . . . 12 ⊢ ((𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp))
10	9	3exp 1116	. . . . . . . . . . 11 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) → (∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp))))
11	10	impcom 411	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) → (∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp)))
12	11	com3l 89	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) → (𝐺 ∈ GrpOp → ((∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) → 𝐺 ∈ MndOp)))
13	7, 12	syl 17	. . . . . . . 8 ⊢ (∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)) → (𝐺 ∈ GrpOp → ((∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) → 𝐺 ∈ MndOp)))
14	4, 13	mpcom 38	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ((∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) → 𝐺 ∈ MndOp))
15	14	expdcom 418	. . . . . 6 ⊢ (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp)))
16	15	a1i 11	. . . . 5 ⊢ (∃𝑤 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑤𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑤) → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp))))
17	16	com13 88	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) → (∃𝑤 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑤𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑤) → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp))))
18	17	3imp 1108	. . 3 ⊢ ((𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑤 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑤𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑤)) → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp))
19	3, 18	syli 39	. 2 ⊢ (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp))
20	19	pm2.43i 52	1 ⊢ (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   × cxp 5517  ran crn 5520  ⟶wf 6320  (class class class)co 7135  GrpOpcgr 28272  MndOpcmndo 35304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-ov 7138  df-grpo 28276  df-ass 35281  df-exid 35283  df-mgmOLD 35287  df-sgrOLD 35299  df-mndo 35305

This theorem is referenced by:  isdrngo2  35396