Theorem grpidval 18639

Description: The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
grpidval.b	⊢ 𝐵 = (Base‘𝐺)
grpidval.p	⊢ + = (+g‘𝐺)
grpidval.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
grpidval	⊢ 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))

Distinct variable groups:   𝑥,𝑒,𝐵   𝑒,𝐺,𝑥

Allowed substitution hints:   + (𝑥,𝑒)   0 (𝑥,𝑒)

Proof of Theorem grpidval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpidval.o	. 2 ⊢ 0 = (0g‘𝐺)
2		fveq2 6876	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpidval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2788	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5	4	eleq2d 2820	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒 ∈ 𝐵))
6		fveq2 6876	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		grpidval.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2788	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9	8	oveqd 7422	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑒(+g‘𝑔)𝑥) = (𝑒 + 𝑥))
10	9	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑒(+g‘𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
11	8	oveqd 7422	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑒) = (𝑥 + 𝑒))
12	11	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
13	10, 12	anbi12d 632	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
14	4, 13	raleqbidv 3325	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
15	5, 14	anbi12d 632	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) ↔ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
16	15	iotabidv 6515	. . . 4 ⊢ (𝑔 = 𝐺 → (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
17		df-0g 17455	. . . 4 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
18		iotaex 6504	. . . 4 ⊢ (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V
19	16, 17, 18	fvmpt 6986	. . 3 ⊢ (𝐺 ∈ V → (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
20		fvprc 6868	. . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅)
21		euex 2576	. . . . . 6 ⊢ (∃!𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → ∃𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
22		n0i 4315	. . . . . . . . 9 ⊢ (𝑒 ∈ 𝐵 → ¬ 𝐵 = ∅)
23		fvprc 6868	. . . . . . . . . 10 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
24	3, 23	eqtrid 2782	. . . . . . . . 9 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
25	22, 24	nsyl2 141	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐵 → 𝐺 ∈ V)
26	25	adantr 480	. . . . . . 7 ⊢ ((𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
27	26	exlimiv 1930	. . . . . 6 ⊢ (∃𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
28	21, 27	syl 17	. . . . 5 ⊢ (∃!𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
29		iotanul 6509	. . . . 5 ⊢ (¬ ∃!𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
30	28, 29	nsyl5 159	. . . 4 ⊢ (¬ 𝐺 ∈ V → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
31	20, 30	eqtr4d 2773	. . 3 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
32	19, 31	pm2.61i 182	. 2 ⊢ (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
33	1, 32	eqtri 2758	1 ⊢ 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567  ∀wral 3051  Vcvv 3459  ∅c0 4308  ℩cio 6482  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  0_gc0g 17453

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-0g 17455

This theorem is referenced by:  grpidpropd  18640  0g0  18642  ismgmid  18643  sgrpidmnd  18717  oppgid  19339  dfur2  20144  oppr0  20309  oppr1  20310  urpropd  33227  opprqus0g  33505