Theorem ringidval 14248

Description: The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
ringidval.g	⊢ 𝐺 = (mulGrp‘𝑅)
ringidval.u	⊢ 1 = (1r‘𝑅)

Assertion

Ref	Expression
ringidval	⊢ 1 = (0g‘𝐺)

Proof of Theorem ringidval

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 5284	. . . . . . 7 ⊢ Rel (0g ∘ mulGrp)
2		df-ur 14246	. . . . . . . 8 ⊢ 1r = (0g ∘ mulGrp)
3	2	releqi 4856	. . . . . . 7 ⊢ (Rel 1r ↔ Rel (0g ∘ mulGrp))
4	1, 3	mpbir 146	. . . . . 6 ⊢ Rel 1r
5		relelfvdm 5725	. . . . . 6 ⊢ ((Rel 1r ∧ 𝑥 ∈ (1r‘𝑅)) → 𝑅 ∈ dom 1r)
6	4, 5	mpan 428	. . . . 5 ⊢ (𝑥 ∈ (1r‘𝑅) → 𝑅 ∈ dom 1r)
7	6	elexd 2835	. . . 4 ⊢ (𝑥 ∈ (1r‘𝑅) → 𝑅 ∈ V)
8		ringidval.u	. . . 4 ⊢ 1 = (1r‘𝑅)
9	7, 8	eleq2s 2333	. . 3 ⊢ (𝑥 ∈ 1 → 𝑅 ∈ V)
10		fn0g 13678	. . . . . . . . . . . . 13 ⊢ 0g Fn V
11		fnrel 5477	. . . . . . . . . . . . 13 ⊢ (0g Fn V → Rel 0g)
12	10, 11	ax-mp 5	. . . . . . . . . . . 12 ⊢ Rel 0g
13		relelfvdm 5725	. . . . . . . . . . . 12 ⊢ ((Rel 0g ∧ 𝑥 ∈ (0g‘𝐺)) → 𝐺 ∈ dom 0g)
14	12, 13	mpan 428	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0g‘𝐺) → 𝐺 ∈ dom 0g)
15	14	elexd 2835	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0g‘𝐺) → 𝐺 ∈ V)
16		eqid 2238	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2238	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
18		eqid 2238	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	16, 17, 18	grpidvalg 13676	. . . . . . . . . 10 ⊢ (𝐺 ∈ V → (0g‘𝐺) = (℩𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧))))
20	15, 19	syl 14	. . . . . . . . 9 ⊢ (𝑥 ∈ (0g‘𝐺) → (0g‘𝐺) = (℩𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧))))
21	20	eleq2d 2308	. . . . . . . 8 ⊢ (𝑥 ∈ (0g‘𝐺) → (𝑥 ∈ (0g‘𝐺) ↔ 𝑥 ∈ (℩𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧)))))
22	21	ibi 176	. . . . . . 7 ⊢ (𝑥 ∈ (0g‘𝐺) → 𝑥 ∈ (℩𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧))))
23		eliotaeu 5364	. . . . . . 7 ⊢ (𝑥 ∈ (℩𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧))) → ∃!𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧)))
24	22, 23	syl 14	. . . . . 6 ⊢ (𝑥 ∈ (0g‘𝐺) → ∃!𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧)))
25		euex 2116	. . . . . 6 ⊢ (∃!𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧)) → ∃𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧)))
26	24, 25	syl 14	. . . . 5 ⊢ (𝑥 ∈ (0g‘𝐺) → ∃𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧)))
27		exsimpl 1670	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) = 𝑧 ∧ (𝑧(+g‘𝐺)𝑦) = 𝑧)) → ∃𝑦 𝑦 ∈ (Base‘𝐺))
28	26, 27	syl 14	. . . 4 ⊢ (𝑥 ∈ (0g‘𝐺) → ∃𝑦 𝑦 ∈ (Base‘𝐺))
29	16	basm 13397	. . . . . . 7 ⊢ (𝑦 ∈ (Base‘𝐺) → ∃𝑤 𝑤 ∈ 𝐺)
30		fnmgp 14202	. . . . . . . . . . 11 ⊢ mulGrp Fn V
31		fnrel 5477	. . . . . . . . . . 11 ⊢ (mulGrp Fn V → Rel mulGrp)
32	30, 31	ax-mp 5	. . . . . . . . . 10 ⊢ Rel mulGrp
33		relelfvdm 5725	. . . . . . . . . 10 ⊢ ((Rel mulGrp ∧ 𝑤 ∈ (mulGrp‘𝑅)) → 𝑅 ∈ dom mulGrp)
34	32, 33	mpan 428	. . . . . . . . 9 ⊢ (𝑤 ∈ (mulGrp‘𝑅) → 𝑅 ∈ dom mulGrp)
35		ringidval.g	. . . . . . . . 9 ⊢ 𝐺 = (mulGrp‘𝑅)
36	34, 35	eleq2s 2333	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐺 → 𝑅 ∈ dom mulGrp)
37	36	exlimiv 1651	. . . . . . 7 ⊢ (∃𝑤 𝑤 ∈ 𝐺 → 𝑅 ∈ dom mulGrp)
38	29, 37	syl 14	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝐺) → 𝑅 ∈ dom mulGrp)
39	38	elexd 2835	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝐺) → 𝑅 ∈ V)
40	39	exlimiv 1651	. . . 4 ⊢ (∃𝑦 𝑦 ∈ (Base‘𝐺) → 𝑅 ∈ V)
41	28, 40	syl 14	. . 3 ⊢ (𝑥 ∈ (0g‘𝐺) → 𝑅 ∈ V)
42	35, 8	ringidvalg 14247	. . . 4 ⊢ (𝑅 ∈ V → 1 = (0g‘𝐺))
43	42	eleq2d 2308	. . 3 ⊢ (𝑅 ∈ V → (𝑥 ∈ 1 ↔ 𝑥 ∈ (0g‘𝐺)))
44	9, 41, 43	pm5.21nii 716	. 2 ⊢ (𝑥 ∈ 1 ↔ 𝑥 ∈ (0g‘𝐺))
45	44	eqriv 2235	1 ⊢ 1 = (0g‘𝐺)

Syntax hints:   ∧ wa 104   = wceq 1402  ∃wex 1545  ∃!weu 2086   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  dom cdm 4772   ∘ ccom 4776  Rel wrel 4777  ℩cio 5333   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  0_gc0g 13593  mulGrpcmgp 14200  1_rcur 14245

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgp 14201  df-ur 14246

This theorem is referenced by:  assamulgscmlem1  15024