Theorem grpidvalg 13249

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 13134	. . 3 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
3		fveq2 5583	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpidval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2257	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	eleq2d 2276	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒 ∈ 𝐵))
7		fveq2 5583	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		grpidval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2257	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 5968	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑒(+g‘𝑔)𝑥) = (𝑒 + 𝑥))
11	10	eqeq1d 2215	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑒(+g‘𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
12	9	oveqd 5968	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑒) = (𝑥 + 𝑒))
13	12	eqeq1d 2215	. . . . . . 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 2719	. . . . 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 5259	. . 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 2784	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
19		df-riota 5906	. . . 4 ⊢ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
20		basfn 12934	. . . . . . 7 ⊢ Base Fn V
21		funfvex 5600	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
22	21	funfni 5381	. . . . . . 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 2293	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
25		riotaexg 5910	. . . . 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 2294	. . 3 ⊢ (𝐺 ∈ 𝑉 → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V)
28	2, 17, 18, 27	fvmptd3 5680	. 2 ⊢ (𝐺 ∈ 𝑉 → (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
29	1, 28	eqtrid 2251	1 ⊢ (𝐺 ∈ 𝑉 → 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  ℩cio 5235   Fn wfn 5271  ‘cfv 5276  ℩crio 5905  (class class class)co 5951  Basecbs 12876  +_gcplusg 12953  0_gc0g 13132

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-fv 5284  df-riota 5906  df-ov 5954  df-inn 9044  df-ndx 12879  df-slot 12880  df-base 12882  df-0g 13134

This theorem is referenced by:  grpidpropdg  13250  0g0  13252  ismgmid  13253  sgrpidmndm  13296  dfur2g  13768  oppr0g  13887  oppr1g  13888