Theorem grpidvalg 13479

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
grpidvalg	⊢ (𝐺 ∈ 𝑉 → 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)   𝑉(𝑥,𝑒)   0 (𝑥,𝑒)

Proof of Theorem grpidvalg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpidval.o	. 2 ⊢ 0 = (0g‘𝐺)
2		df-0g 13364	. . 3 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
3		fveq2 5642	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpidval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2281	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	eleq2d 2300	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒 ∈ 𝐵))
7		fveq2 5642	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		grpidval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2281	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 6040	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑒(+g‘𝑔)𝑥) = (𝑒 + 𝑥))
11	10	eqeq1d 2239	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑒(+g‘𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
12	9	oveqd 6040	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑒) = (𝑥 + 𝑒))
13	12	eqeq1d 2239	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
14	11, 13	anbi12d 473	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
15	5, 14	raleqbidv 2745	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
16	6, 15	anbi12d 473	. . . 4 ⊢ (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))))
17	16	iotabidv 5311	. . 3 ⊢ (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))))
18		elex 2813	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
19		df-riota 5976	. . . 4 ⊢ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
20		basfn 13164	. . . . . . 7 ⊢ Base Fn V
21		funfvex 5659	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
22	21	funfni 5434	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
23	20, 18, 22	sylancr 414	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
24	4, 23	eqeltrid 2317	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
25		riotaexg 5980	. . . . 5 ⊢ (𝐵 ∈ V → (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ V)
26	24, 25	syl 14	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ V)
27	19, 26	eqeltrrid 2318	. . 3 ⊢ (𝐺 ∈ 𝑉 → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V)
28	2, 17, 18, 27	fvmptd3 5743	. 2 ⊢ (𝐺 ∈ 𝑉 → (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
29	1, 28	eqtrid 2275	1 ⊢ (𝐺 ∈ 𝑉 → 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  ∀wral 2509  Vcvv 2801  ℩cio 5286   Fn wfn 5323  ‘cfv 5328  ℩crio 5975  (class class class)co 6023  Basecbs 13105  +_gcplusg 13183  0_gc0g 13362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336  df-riota 5976  df-ov 6026  df-inn 9149  df-ndx 13108  df-slot 13109  df-base 13111  df-0g 13364

This theorem is referenced by:  grpidpropdg  13480  0g0  13482  ismgmid  13483  sgrpidmndm  13526  dfur2g  13999  oppr0g  14118  oppr1g  14119