Theorem ismgmid2 18630

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 = (0g‘𝐺)
ismgmid.p	⊢ + = (+g‘𝐺)
ismgmid2.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)
ismgmid2.l	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥)
ismgmid2.r	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥)

Assertion

Ref	Expression
ismgmid2	⊢ (𝜑 → 𝑈 = 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.

Step	Hyp	Ref	Expression
1		ismgmid2.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
2		ismgmid2.l	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥)
3		ismgmid2.r	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥)
4	2, 3	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))
5	4	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥))
6		ismgmid.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
7		ismgmid.o	. . . 4 ⊢ 0 = (0g‘𝐺)
8		ismgmid.p	. . . 4 ⊢ + = (+g‘𝐺)
9		oveq1 7368	. . . . . . . 8 ⊢ (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥))
10	9	eqeq1d 2739	. . . . . . 7 ⊢ (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥))
11	10	ovanraleqv 7385	. . . . . 6 ⊢ (𝑒 = 𝑈 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
12	11	rspcev 3565	. . . . 5 ⊢ ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
13	1, 5, 12	syl2anc 585	. . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
14	6, 7, 8, 13	ismgmid 18627	. . 3 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
15	1, 5, 14	mpbi2and 713	. 2 ⊢ (𝜑 → 0 = 𝑈)
16	15	eqcomd 2743	1 ⊢ (𝜑 → 𝑈 = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  +_gcplusg 17214  0_gc0g 17396

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-riota 7318  df-ov 7364  df-0g 17398

This theorem is referenced by:  lidrididd  18632  grpidd  18633  submnd0  18725  xpsmnd0  18740  mnd1id  18742  frmd0  18822  efmndid  18850  pwmndid  18901  mhmid  19033  cnaddid  19839  ringidss  20252  xpsring1d  20307  xrs10  21434  rloccring  33349  idlsrg0g  33584