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Theorem ismgmid2 17183
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 554 . . . 4 ((𝜑𝑥𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))
54ralrimiva 2965 . . 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 6612 . . . . . . . . 9 (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥))
109eqeq1d 2628 . . . . . . . 8 (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥))
11 oveq2 6613 . . . . . . . . 9 (𝑒 = 𝑈 → (𝑥 + 𝑒) = (𝑥 + 𝑈))
1211eqeq1d 2628 . . . . . . . 8 (𝑒 = 𝑈 → ((𝑥 + 𝑒) = 𝑥 ↔ (𝑥 + 𝑈) = 𝑥))
1310, 12anbi12d 746 . . . . . . 7 (𝑒 = 𝑈 → (((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
1413ralbidv 2985 . . . . . 6 (𝑒 = 𝑈 → (∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
1514rspcev 3300 . . . . 5 ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
161, 5, 15syl2anc 692 . . . 4 (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
176, 7, 8, 16ismgmid 17180 . . 3 (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
181, 5, 17mpbi2and 955 . 2 (𝜑0 = 𝑈)
1918eqcomd 2632 1 (𝜑𝑈 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-riota 6566  df-ov 6608  df-0g 16018
This theorem is referenced by:  grpidd  17184  submnd0  17236  mnd1id  17248  frmd0  17313  mhmid  17452  cnaddid  18189  ringidss  18493  xrs10  19699
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