Theorem issrgid 12957

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 2175	. . 3 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
2		eqid 2175	. . 3 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
3		eqid 2175	. . 3 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
4		srgidm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5		srgidm.t	. . . . . 6 ⊢ · = (.r‘𝑅)
6	4, 5	srgideu 12948	. . . . 5 ⊢ (𝑅 ∈ SRing → ∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑦) = 𝑥))
7		reurex 2688	. . . . 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 2175	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
10	9, 4	mgpbasg 12930	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝐵 = (Base‘(mulGrp‘𝑅)))
11	9, 5	mgpplusgg 12929	. . . . . . . . 9 ⊢ (𝑅 ∈ SRing → · = (+g‘(mulGrp‘𝑅)))
12	11	oveqd 5882	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → (𝑦 · 𝑥) = (𝑦(+g‘(mulGrp‘𝑅))𝑥))
13	12	eqeq1d 2184	. . . . . . 7 ⊢ (𝑅 ∈ SRing → ((𝑦 · 𝑥) = 𝑥 ↔ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑥))
14	11	oveqd 5882	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → (𝑥 · 𝑦) = (𝑥(+g‘(mulGrp‘𝑅))𝑦))
15	14	eqeq1d 2184	. . . . . . 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 2682	. . . . 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 2683	. . . 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 12661	. 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 2245	. . 3 ⊢ (𝑅 ∈ SRing → (𝐼 ∈ 𝐵 ↔ 𝐼 ∈ (Base‘(mulGrp‘𝑅))))
22	11	oveqd 5882	. . . . . 6 ⊢ (𝑅 ∈ SRing → (𝐼 · 𝑥) = (𝐼(+g‘(mulGrp‘𝑅))𝑥))
23	22	eqeq1d 2184	. . . . 5 ⊢ (𝑅 ∈ SRing → ((𝐼 · 𝑥) = 𝑥 ↔ (𝐼(+g‘(mulGrp‘𝑅))𝑥) = 𝑥))
24	11	oveqd 5882	. . . . . 6 ⊢ (𝑅 ∈ SRing → (𝑥 · 𝐼) = (𝑥(+g‘(mulGrp‘𝑅))𝐼))
25	24	eqeq1d 2184	. . . . 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 2682	. . 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 12937	. . 3 ⊢ (𝑅 ∈ SRing → 1 = (0g‘(mulGrp‘𝑅)))
31	30	eqeq1d 2184	. 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 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  ∃!wreu 2455  ‘cfv 5208  (class class class)co 5865  Basecbs 12428  +_gcplusg 12492  ._rcmulr 12493  0_gc0g 12626  mulGrpcmgp 12925  1_rcur 12935  SRingcsrg 12939

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12431  df-slot 12432  df-base 12434  df-sets 12435  df-plusg 12505  df-mulr 12506  df-0g 12628  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-mgp 12926  df-ur 12936  df-srg 12940

This theorem is referenced by: (None)