Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exidu1 Structured version   Visualization version   GIF version

Theorem exidu1 36014
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 ) → ∃!𝑢𝑋𝑥𝑋 ((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.
StepHypRef Expression
1 exidu1.1 . . 3 𝑋 = ran 𝐺
21isexid2 36013 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
3 simpl 483 . . . . . . 7 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
43ralimi 3087 . . . . . 6 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
5 oveq2 7283 . . . . . . . 8 (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
6 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
75, 6eqeq12d 2754 . . . . . . 7 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
87rspcv 3557 . . . . . 6 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
94, 8syl5 34 . . . . 5 (𝑦𝑋 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑦) = 𝑦))
10 simpr 485 . . . . . . 7 (((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑥𝐺𝑦) = 𝑥)
1110ralimi 3087 . . . . . 6 (∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → ∀𝑥𝑋 (𝑥𝐺𝑦) = 𝑥)
12 oveq1 7282 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥𝐺𝑦) = (𝑢𝐺𝑦))
13 id 22 . . . . . . . 8 (𝑥 = 𝑢𝑥 = 𝑢)
1412, 13eqeq12d 2754 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥𝐺𝑦) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑢))
1514rspcv 3557 . . . . . 6 (𝑢𝑋 → (∀𝑥𝑋 (𝑥𝐺𝑦) = 𝑥 → (𝑢𝐺𝑦) = 𝑢))
1611, 15syl5 34 . . . . 5 (𝑢𝑋 → (∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑢𝐺𝑦) = 𝑢))
179, 16im2anan9r 621 . . . 4 ((𝑢𝑋𝑦𝑋) → ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → ((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢)))
18 eqtr2 2762 . . . . 5 (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑦 = 𝑢)
1918equcomd 2022 . . . 4 (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑢 = 𝑦)
2017, 19syl6 35 . . 3 ((𝑢𝑋𝑦𝑋) → ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦))
2120rgen2 3120 . 2 𝑢𝑋𝑦𝑋 ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)
22 oveq1 7282 . . . . 5 (𝑢 = 𝑦 → (𝑢𝐺𝑥) = (𝑦𝐺𝑥))
2322eqeq1d 2740 . . . 4 (𝑢 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑦𝐺𝑥) = 𝑥))
2423ovanraleqv 7299 . . 3 (𝑢 = 𝑦 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)))
2524reu4 3666 . 2 (∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑢𝑋𝑦𝑋 ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)))
262, 21, 25sylanblrc 590 1 (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  ∃!wreu 3066  cin 3886  ran crn 5590  (class class class)co 7275   ExId cexid 36002  Magmacmagm 36006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-exid 36003  df-mgmOLD 36007
This theorem is referenced by:  iorlid  36016  cmpidelt  36017  exidresid  36037
  Copyright terms: Public domain W3C validator