Theorem grpidval 17260

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 6191	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpidval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	syl6eqr 2674	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5	4	eleq2d 2687	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒 ∈ 𝐵))
6		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		grpidval.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
8	6, 7	syl6eqr 2674	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9	8	oveqd 6667	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑒(+g‘𝑔)𝑥) = (𝑒 + 𝑥))
10	9	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑒(+g‘𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
11	8	oveqd 6667	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑒) = (𝑥 + 𝑒))
12	11	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
13	10, 12	anbi12d 747	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
14	4, 13	raleqbidv 3152	. . . . . 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 747	. . . . 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 5872	. . . 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 16102	. . . 4 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
18		iotaex 5868	. . . 4 ⊢ (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V
19	16, 17, 18	fvmpt 6282	. . 3 ⊢ (𝐺 ∈ V → (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
20		fvprc 6185	. . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅)
21		euex 2494	. . . . . . 7 ⊢ (∃!𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → ∃𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
22		n0i 3920	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐵 → ¬ 𝐵 = ∅)
23		fvprc 6185	. . . . . . . . . . 11 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
24	3, 23	syl5eq 2668	. . . . . . . . . 10 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
25	22, 24	nsyl2 142	. . . . . . . . 9 ⊢ (𝑒 ∈ 𝐵 → 𝐺 ∈ V)
26	25	adantr 481	. . . . . . . 8 ⊢ ((𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
27	26	exlimiv 1858	. . . . . . 7 ⊢ (∃𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
28	21, 27	syl 17	. . . . . 6 ⊢ (∃!𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
29	28	con3i 150	. . . . 5 ⊢ (¬ 𝐺 ∈ V → ¬ ∃!𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
30		iotanul 5866	. . . . 5 ⊢ (¬ ∃!𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
31	29, 30	syl 17	. . . 4 ⊢ (¬ 𝐺 ∈ V → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
32	20, 31	eqtr4d 2659	. . 3 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
33	19, 32	pm2.61i 176	. 2 ⊢ (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
34	1, 33	eqtri 2644	1 ⊢ 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∀wral 2912  Vcvv 3200  ∅c0 3915  ℩cio 5849  ‘cfv 5888  (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-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-ov 6653  df-0g 16102

This theorem is referenced by:  grpidpropd  17261  0g0  17263  ismgmid  17264  oppgid  17786  dfur2  18504  oppr0  18633  oppr1  18634