Theorem exidu1 33655

Description: Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
exidu1.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
exidu1	⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Distinct variable groups:   𝑢,𝐺,𝑥   𝑢,𝑋,𝑥

Proof of Theorem exidu1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exidu1.1	. . 3 ⊢ 𝑋 = ran 𝐺
2	1	isexid2 33654	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
3		simpl 473	. . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
4	3	ralimi 2952	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
5		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
6		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
7	5, 6	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
8	7	rspcv 3305	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
9	4, 8	syl5 34	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑦) = 𝑦))
10		simpr 477	. . . . . . . 8 ⊢ (((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑥𝐺𝑦) = 𝑥)
11	10	ralimi 2952	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑥𝐺𝑦) = 𝑥)
12		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥𝐺𝑦) = (𝑢𝐺𝑦))
13		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
14	12, 13	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥𝐺𝑦) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑢))
15	14	rspcv 3305	. . . . . . 7 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑥𝐺𝑦) = 𝑥 → (𝑢𝐺𝑦) = 𝑢))
16	11, 15	syl5 34	. . . . . 6 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑢𝐺𝑦) = 𝑢))
17	9, 16	im2anan9r 881	. . . . 5 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → ((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢)))
18		eqtr2 2642	. . . . . 6 ⊢ (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑦 = 𝑢)
19	18	eqcomd 2628	. . . . 5 ⊢ (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑢 = 𝑦)
20	17, 19	syl6 35	. . . 4 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦))
21	20	rgen2a 2977	. . 3 ⊢ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)
22	21	a1i 11	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦))
23		oveq1 6657	. . . . . 6 ⊢ (𝑢 = 𝑦 → (𝑢𝐺𝑥) = (𝑦𝐺𝑥))
24	23	eqeq1d 2624	. . . . 5 ⊢ (𝑢 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑦𝐺𝑥) = 𝑥))
25		oveq2 6658	. . . . . 6 ⊢ (𝑢 = 𝑦 → (𝑥𝐺𝑢) = (𝑥𝐺𝑦))
26	25	eqeq1d 2624	. . . . 5 ⊢ (𝑢 = 𝑦 → ((𝑥𝐺𝑢) = 𝑥 ↔ (𝑥𝐺𝑦) = 𝑥))
27	24, 26	anbi12d 747	. . . 4 ⊢ (𝑢 = 𝑦 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)))
28	27	ralbidv 2986	. . 3 ⊢ (𝑢 = 𝑦 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)))
29	28	reu4 3400	. 2 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)))
30	2, 22, 29	sylanbrc 698	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914   ∩ cin 3573  ran crn 5115  (class class class)co 6650   ExId cexid 33643  Magmacmagm 33647

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-reu 2919  df-rmo 2920  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-exid 33644  df-mgmOLD 33648

This theorem is referenced by:  iorlid  33657  cmpidelt  33658  exidresid  33678