Theorem exidu1 38429

Description: Uniqueness 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 38428	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
3		simpl 487	. . . . . . 7 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
4	3	ralimi 3108	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
5		oveq2 7419	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
6		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
7	5, 6	eqeq12d 2785	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
8	7	rspcv 3586	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
9	4, 8	syl5 35	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑦) = 𝑦))
10		simpr 489	. . . . . . 7 ⊢ (((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑥𝐺𝑦) = 𝑥)
11	10	ralimi 3108	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑥𝐺𝑦) = 𝑥)
12		oveq1 7418	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥𝐺𝑦) = (𝑢𝐺𝑦))
13		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
14	12, 13	eqeq12d 2785	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝑥𝐺𝑦) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑢))
15	14	rspcv 3586	. . . . . 6 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑥𝐺𝑦) = 𝑥 → (𝑢𝐺𝑦) = 𝑢))
16	11, 15	syl5 35	. . . . 5 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑢𝐺𝑦) = 𝑢))
17	9, 16	im2anan9r 632	. . . 4 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → ((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢)))
18		eqtr2 2790	. . . . 5 ⊢ (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑦 = 𝑢)
19	18	equcomd 2046	. . . 4 ⊢ (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑢 = 𝑦)
20	17, 19	syl6 36	. . 3 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦))
21	20	rgen2 3211	. 2 ⊢ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)
22		oveq1 7418	. . . . 5 ⊢ (𝑢 = 𝑦 → (𝑢𝐺𝑥) = (𝑦𝐺𝑥))
23	22	eqeq1d 2771	. . . 4 ⊢ (𝑢 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑦𝐺𝑥) = 𝑥))
24	23	ovanraleqv 7435	. . 3 ⊢ (𝑢 = 𝑦 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)))
25	24	reu4 3703	. 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)))
26	2, 21, 25	sylanblrc 601	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374   ∩ cin 3912  ran crn 5663  (class class class)co 7411   ExId cexid 38417  Magmacmagm 38421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7414  df-exid 38418  df-mgmOLD 38422

This theorem is referenced by:  iorlid  38431  cmpidelt  38432  exidresid  38452