Theorem grpomndo 35770

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 2737	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
2	1	isgrpo 28578	. . . 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 28589	. . . . . . . 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 3083	. . . . . . . . . 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 3166	. . . . . . . . 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 35769	. . . . . . . . . . . . 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 1121	. . . . . . . . . . 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 1113	. . 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 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062   × cxp 5549  ran crn 5552  ⟶wf 6376  (class class class)co 7213  GrpOpcgr 28570  MndOpcmndo 35761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-ov 7216  df-grpo 28574  df-ass 35738  df-exid 35740  df-mgmOLD 35744  df-sgrOLD 35756  df-mndo 35762

This theorem is referenced by:  isdrngo2  35853