Theorem issrgid 13480

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 2193	. . 3 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
2		eqid 2193	. . 3 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
3		eqid 2193	. . 3 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
4		srgidm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5		srgidm.t	. . . . . 6 ⊢ · = (.r‘𝑅)
6	4, 5	srgideu 13471	. . . . 5 ⊢ (𝑅 ∈ SRing → ∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥))
7		reurex 2712	. . . . 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 2193	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
10	9, 4	mgpbasg 13425	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝐵 = (Base‘(mulGrp‘𝑅)))
11	9, 5	mgpplusgg 13423	. . . . . . . . 9 ⊢ (𝑅 ∈ SRing → · = (+g‘(mulGrp‘𝑅)))
12	11	oveqd 5936	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → (𝑦 · 𝑥) = (𝑦(+g‘(mulGrp‘𝑅))𝑥))
13	12	eqeq1d 2202	. . . . . . 7 ⊢ (𝑅 ∈ SRing → ((𝑦 · 𝑥) = 𝑥 ↔ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑥))
14	11	oveqd 5936	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → (𝑥 · 𝑦) = (𝑥(+g‘(mulGrp‘𝑅))𝑦))
15	14	eqeq1d 2202	. . . . . . 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 2706	. . . . 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 2707	. . . 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 12963	. 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 2263	. . 3 ⊢ (𝑅 ∈ SRing → (𝐼 ∈ 𝐵 ↔ 𝐼 ∈ (Base‘(mulGrp‘𝑅))))
22	11	oveqd 5936	. . . . . 6 ⊢ (𝑅 ∈ SRing → (𝐼 · 𝑥) = (𝐼(+g‘(mulGrp‘𝑅))𝑥))
23	22	eqeq1d 2202	. . . . 5 ⊢ (𝑅 ∈ SRing → ((𝐼 · 𝑥) = 𝑥 ↔ (𝐼(+g‘(mulGrp‘𝑅))𝑥) = 𝑥))
24	11	oveqd 5936	. . . . . 6 ⊢ (𝑅 ∈ SRing → (𝑥 · 𝐼) = (𝑥(+g‘(mulGrp‘𝑅))𝐼))
25	24	eqeq1d 2202	. . . . 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 2706	. . 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 13460	. . 3 ⊢ (𝑅 ∈ SRing → 1 = (0g‘(mulGrp‘𝑅)))
31	30	eqeq1d 2202	. 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 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∃!wreu 2474  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  +_gcplusg 12698  ._rcmulr 12699  0_gc0g 12870  mulGrpcmgp 13419  1_rcur 13458  SRingcsrg 13462

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mgp 13420  df-ur 13459  df-srg 13463

This theorem is referenced by: (None)