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Theorem ismgmid2 17090
 Description: Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
ismgmid.b 𝐵 = (Base‘𝐺)
ismgmid.o 0 = (0g𝐺)
ismgmid.p + = (+g𝐺)
ismgmid2.u (𝜑𝑈𝐵)
ismgmid2.l ((𝜑𝑥𝐵) → (𝑈 + 𝑥) = 𝑥)
ismgmid2.r ((𝜑𝑥𝐵) → (𝑥 + 𝑈) = 𝑥)
Assertion
Ref Expression
ismgmid2 (𝜑𝑈 = 0 )
Distinct variable groups:   𝑥, +   𝑥, 0   𝑥,𝐵   𝑥,𝐺   𝑥,𝑈   𝜑,𝑥

Proof of Theorem ismgmid2
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 ismgmid2.u . . 3 (𝜑𝑈𝐵)
2 ismgmid2.l . . . . 5 ((𝜑𝑥𝐵) → (𝑈 + 𝑥) = 𝑥)
3 ismgmid2.r . . . . 5 ((𝜑𝑥𝐵) → (𝑥 + 𝑈) = 𝑥)
42, 3jca 553 . . . 4 ((𝜑𝑥𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))
54ralrimiva 2949 . . 3 (𝜑 → ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))
6 ismgmid.b . . . 4 𝐵 = (Base‘𝐺)
7 ismgmid.o . . . 4 0 = (0g𝐺)
8 ismgmid.p . . . 4 + = (+g𝐺)
9 oveq1 6556 . . . . . . . . 9 (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥))
109eqeq1d 2612 . . . . . . . 8 (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥))
11 oveq2 6557 . . . . . . . . 9 (𝑒 = 𝑈 → (𝑥 + 𝑒) = (𝑥 + 𝑈))
1211eqeq1d 2612 . . . . . . . 8 (𝑒 = 𝑈 → ((𝑥 + 𝑒) = 𝑥 ↔ (𝑥 + 𝑈) = 𝑥))
1310, 12anbi12d 743 . . . . . . 7 (𝑒 = 𝑈 → (((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
1413ralbidv 2969 . . . . . 6 (𝑒 = 𝑈 → (∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
1514rspcev 3282 . . . . 5 ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
161, 5, 15syl2anc 691 . . . 4 (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
176, 7, 8, 16ismgmid 17087 . . 3 (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
181, 5, 17mpbi2and 958 . 2 (𝜑0 = 𝑈)
1918eqcomd 2616 1 (𝜑𝑈 = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925 This theorem is referenced by:  grpidd  17091  submnd0  17143  mnd1id  17155  frmd0  17220  mhmid  17359  cnaddid  18096  ringidss  18400  xrs10  19604
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