Theorem ismgmid 17264

Description: The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
ismgmid.b	⊢ 𝐵 = (Base‘𝐺)
ismgmid.o	⊢ 0 = (0g‘𝐺)
ismgmid.p	⊢ + = (+g‘𝐺)
mgmidcl.e	⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

Assertion

Ref	Expression
ismgmid	⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))

Distinct variable groups:   𝑥,𝑒, +   0 ,𝑒,𝑥   𝐵,𝑒,𝑥   𝑒,𝐺,𝑥   𝑈,𝑒,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑒)

Proof of Theorem ismgmid

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑈 ∈ 𝐵 → 𝑈 ∈ 𝐵)
2		mgmidcl.e	. . . . 5 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
3		mgmidmo 17259	. . . . . 6 ⊢ ∃*𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → ∃*𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
5		reu5 3159	. . . . 5 ⊢ (∃!𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ (∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ∧ ∃*𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
6	2, 4, 5	sylanbrc 698	. . . 4 ⊢ (𝜑 → ∃!𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
7		oveq1 6657	. . . . . . . 8 ⊢ (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥))
8	7	eqeq1d 2624	. . . . . . 7 ⊢ (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥))
9		oveq2 6658	. . . . . . . 8 ⊢ (𝑒 = 𝑈 → (𝑥 + 𝑒) = (𝑥 + 𝑈))
10	9	eqeq1d 2624	. . . . . . 7 ⊢ (𝑒 = 𝑈 → ((𝑥 + 𝑒) = 𝑥 ↔ (𝑥 + 𝑈) = 𝑥))
11	8, 10	anbi12d 747	. . . . . 6 ⊢ (𝑒 = 𝑈 → (((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
12	11	ralbidv 2986	. . . . 5 ⊢ (𝑒 = 𝑈 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
13	12	riota2 6633	. . . 4 ⊢ ((𝑈 ∈ 𝐵 ∧ ∃!𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → (∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥) ↔ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈))
14	1, 6, 13	syl2anr 495	. . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥) ↔ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈))
15	14	pm5.32da 673	. 2 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ (𝑈 ∈ 𝐵 ∧ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈)))
16		riotacl 6625	. . . . 5 ⊢ (∃!𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ 𝐵)
17	6, 16	syl 17	. . . 4 ⊢ (𝜑 → (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ 𝐵)
18		eleq1 2689	. . . 4 ⊢ ((℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈 → ((℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ 𝐵 ↔ 𝑈 ∈ 𝐵))
19	17, 18	syl5ibcom 235	. . 3 ⊢ (𝜑 → ((℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈 → 𝑈 ∈ 𝐵))
20	19	pm4.71rd 667	. 2 ⊢ (𝜑 → ((℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈 ↔ (𝑈 ∈ 𝐵 ∧ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈)))
21		df-riota 6611	. . . . 5 ⊢ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
22		ismgmid.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
23		ismgmid.p	. . . . . 6 ⊢ + = (+g‘𝐺)
24		ismgmid.o	. . . . . 6 ⊢ 0 = (0g‘𝐺)
25	22, 23, 24	grpidval 17260	. . . . 5 ⊢ 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
26	21, 25	eqtr4i 2647	. . . 4 ⊢ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 0
27	26	eqeq1i 2627	. . 3 ⊢ ((℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈 ↔ 0 = 𝑈)
28	27	a1i 11	. 2 ⊢ (𝜑 → ((℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈 ↔ 0 = 𝑈))
29	15, 20, 28	3bitr2d 296	1 ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  ∃*wrmo 2915  ℩cio 5849  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102

This theorem is referenced by:  mgmidcl  17265  mgmlrid  17266  ismgmid2  17267  mgmidsssn0  17269  prds0g  17324  issrgid  18523  isringid  18573  signsw0g  30633