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Theorem ismgmid2 18681
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 3146 . . 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 2739 . . . . . . 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 18678 . . 3 (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
151, 5, 14mpbi2and 712 . 2 (𝜑0 = 𝑈)
1615eqcomd 2743 1 (𝜑𝑈 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486
This theorem is referenced by:  lidrididd  18683  grpidd  18684  submnd0  18776  xpsmnd0  18791  mnd1id  18793  frmd0  18873  efmndid  18901  pwmndid  18949  mhmid  19081  cnaddid  19888  ringidss  20274  xpsring1d  20330  xrs10  21423  rloccring  33274  idlsrg0g  33534
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