Theorem issrgid 13984

Description: Properties showing that an element 𝐼 is the unity element of a semiring. (Contributed by NM, 7-Aug-2013.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
srgidm.b	⊢ 𝐵 = (Base‘𝑅)
srgidm.t	⊢ · = (.r‘𝑅)
srgidm.u	⊢ 1 = (1r‘𝑅)

Assertion

Ref	Expression
issrgid	⊢ (𝑅 ∈ SRing → ((𝐼 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐼   𝑥,𝑅   𝑥, ·   𝑥, 1

Proof of Theorem issrgid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. . 3 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
2		eqid 2229	. . 3 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
3		eqid 2229	. . 3 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
4		srgidm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5		srgidm.t	. . . . . 6 ⊢ · = (.r‘𝑅)
6	4, 5	srgideu 13975	. . . . 5 ⊢ (𝑅 ∈ SRing → ∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥))
7		reurex 2750	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥) → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥))
8	6, 7	syl 14	. . . 4 ⊢ (𝑅 ∈ SRing → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥))
9		eqid 2229	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
10	9, 4	mgpbasg 13929	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝐵 = (Base‘(mulGrp‘𝑅)))
11	9, 5	mgpplusgg 13927	. . . . . . . . 9 ⊢ (𝑅 ∈ SRing → · = (+g‘(mulGrp‘𝑅)))
12	11	oveqd 6030	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → (𝑦 · 𝑥) = (𝑦(+g‘(mulGrp‘𝑅))𝑥))
13	12	eqeq1d 2238	. . . . . . 7 ⊢ (𝑅 ∈ SRing → ((𝑦 · 𝑥) = 𝑥 ↔ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑥))
14	11	oveqd 6030	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → (𝑥 · 𝑦) = (𝑥(+g‘(mulGrp‘𝑅))𝑦))
15	14	eqeq1d 2238	. . . . . . 7 ⊢ (𝑅 ∈ SRing → ((𝑥 · 𝑦) = 𝑥 ↔ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑥))
16	13, 15	anbi12d 473	. . . . . 6 ⊢ (𝑅 ∈ SRing → (((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥) ↔ ((𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑥)))
17	10, 16	raleqbidv 2744	. . . . 5 ⊢ (𝑅 ∈ SRing → (∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑥)))
18	10, 17	rexeqbidv 2745	. . . 4 ⊢ (𝑅 ∈ SRing → (∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥) ↔ ∃𝑦 ∈ (Base‘(mulGrp‘𝑅))∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑥)))
19	8, 18	mpbid 147	. . 3 ⊢ (𝑅 ∈ SRing → ∃𝑦 ∈ (Base‘(mulGrp‘𝑅))∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑥))
20	1, 2, 3, 19	ismgmid 13450	. 2 ⊢ (𝑅 ∈ SRing → ((𝐼 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝐼(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝐼) = 𝑥)) ↔ (0g‘(mulGrp‘𝑅)) = 𝐼))
21	10	eleq2d 2299	. . 3 ⊢ (𝑅 ∈ SRing → (𝐼 ∈ 𝐵 ↔ 𝐼 ∈ (Base‘(mulGrp‘𝑅))))
22	11	oveqd 6030	. . . . . 6 ⊢ (𝑅 ∈ SRing → (𝐼 · 𝑥) = (𝐼(+g‘(mulGrp‘𝑅))𝑥))
23	22	eqeq1d 2238	. . . . 5 ⊢ (𝑅 ∈ SRing → ((𝐼 · 𝑥) = 𝑥 ↔ (𝐼(+g‘(mulGrp‘𝑅))𝑥) = 𝑥))
24	11	oveqd 6030	. . . . . 6 ⊢ (𝑅 ∈ SRing → (𝑥 · 𝐼) = (𝑥(+g‘(mulGrp‘𝑅))𝐼))
25	24	eqeq1d 2238	. . . . 5 ⊢ (𝑅 ∈ SRing → ((𝑥 · 𝐼) = 𝑥 ↔ (𝑥(+g‘(mulGrp‘𝑅))𝐼) = 𝑥))
26	23, 25	anbi12d 473	. . . 4 ⊢ (𝑅 ∈ SRing → (((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥) ↔ ((𝐼(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝐼) = 𝑥)))
27	10, 26	raleqbidv 2744	. . 3 ⊢ (𝑅 ∈ SRing → (∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥) ↔ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝐼(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝐼) = 𝑥)))
28	21, 27	anbi12d 473	. 2 ⊢ (𝑅 ∈ SRing → ((𝐼 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ (𝐼 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝐼(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝐼) = 𝑥))))
29		srgidm.u	. . . 4 ⊢ 1 = (1r‘𝑅)
30	9, 29	ringidvalg 13964	. . 3 ⊢ (𝑅 ∈ SRing → 1 = (0g‘(mulGrp‘𝑅)))
31	30	eqeq1d 2238	. 2 ⊢ (𝑅 ∈ SRing → ( 1 = 𝐼 ↔ (0g‘(mulGrp‘𝑅)) = 𝐼))
32	20, 28, 31	3bitr4d 220	1 ⊢ (𝑅 ∈ SRing → ((𝐼 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  +_gcplusg 13150  ._rcmulr 13151  0_gc0g 13329  mulGrpcmgp 13923  1_rcur 13962  SRingcsrg 13966

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mgp 13924  df-ur 13963  df-srg 13967

This theorem is referenced by: (None)