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Theorem ismgmid2 17890
 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 515 . . . 4 ((𝜑𝑥𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))
54ralrimiva 3149 . . 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 7152 . . . . . . . 8 (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥))
109eqeq1d 2800 . . . . . . 7 (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥))
1110ovanraleqv 7169 . . . . . 6 (𝑒 = 𝑈 → (∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
1211rspcev 3572 . . . . 5 ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
131, 5, 12syl2anc 587 . . . 4 (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
146, 7, 8, 13ismgmid 17887 . . 3 (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
151, 5, 14mpbi2and 711 . 2 (𝜑0 = 𝑈)
1615eqcomd 2804 1 (𝜑𝑈 = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  +gcplusg 16577  0gc0g 16725 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fv 6340  df-riota 7103  df-ov 7148  df-0g 16727 This theorem is referenced by:  lidrididd  17892  grpidd  17893  submnd0  17952  mnd1id  17965  frmd0  18037  efmndid  18065  pwmndid  18113  mhmid  18233  cnaddid  19004  ringidss  19344  xrs10  20151  idlsrg0g  31120
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