Theorem grpomndo 33674

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 2622	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
2	1	isgrpo 27351	. . . 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 219	. . 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 27362	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)))
5		simpl 473	. . . . . . . . . . 11 ⊢ ((((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)) → ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))
6	5	ralimi 2952	. . . . . . . . . 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 3011	. . . . . . . . 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 33673	. . . . . . . . . . . . 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 240	. . . . . . . . . . . 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 1264	. . . . . . . . . . 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 446	. . . . . . . . . 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 455	. . . . . 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 1256	. . 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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   × cxp 5112  ran crn 5115  ⟶wf 5884  (class class class)co 6650  GrpOpcgr 27343  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-grpo 27347  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666

This theorem is referenced by:  isdrngo2  33757