Theorem grpomndo 36334

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 2736	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
2	1	isgrpo 29439	. . . 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 228	. . 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 29450	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)))
5		simpl 483	. . . . . . . . . . 11 ⊢ ((((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ∧ ∃𝑤 ∈ ran 𝐺((𝑤𝐺𝑦) = 𝑥 ∧ (𝑦𝐺𝑤) = 𝑥)) → ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))
6	5	ralimi 3086	. . . . . . . . . 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 3087	. . . . . . . . 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 36333	. . . . . . . . . . . . 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 249	. . . . . . . . . . . 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 1119	. . . . . . . . . . 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 408	. . . . . . . . . 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 415	. . . . . 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 1111	. . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   × cxp 5631  ran crn 5634  ⟶wf 6492  (class class class)co 7357  GrpOpcgr 29431  MndOpcmndo 36325

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-ov 7360  df-grpo 29435  df-ass 36302  df-exid 36304  df-mgmOLD 36308  df-sgrOLD 36320  df-mndo 36326

This theorem is referenced by:  isdrngo2  36417