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Theorem ismgmid2 18694
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 511 . . . 4 ((𝜑𝑥𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))
54ralrimiva 3144 . . 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 7438 . . . . . . . 8 (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥))
109eqeq1d 2737 . . . . . . 7 (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥))
1110ovanraleqv 7455 . . . . . 6 (𝑒 = 𝑈 → (∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
1211rspcev 3622 . . . . 5 ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
131, 5, 12syl2anc 584 . . . 4 (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
146, 7, 8, 13ismgmid 18691 . . 3 (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
151, 5, 14mpbi2and 712 . 2 (𝜑0 = 𝑈)
1615eqcomd 2741 1 (𝜑𝑈 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488
This theorem is referenced by:  lidrididd  18696  grpidd  18697  submnd0  18789  xpsmnd0  18804  mnd1id  18806  frmd0  18886  efmndid  18914  pwmndid  18962  mhmid  19094  cnaddid  19903  ringidss  20291  xpsring1d  20347  xrs10  21441  rloccring  33257  idlsrg0g  33514
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