Theorem rngoueqz 37969

Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20497 instead. In a unital ring the zero equals the ring unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)

Hypotheses

Ref	Expression
uznzr.1	⊢ 𝐺 = (1st ‘𝑅)
uznzr.2	⊢ 𝐻 = (2nd ‘𝑅)
uznzr.3	⊢ 𝑍 = (GId‘𝐺)
uznzr.4	⊢ 𝑈 = (GId‘𝐻)
uznzr.5	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngoueqz	⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))

Proof of Theorem rngoueqz

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uznzr.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		uznzr.5	. . . 4 ⊢ 𝑋 = ran 𝐺
3		uznzr.3	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	rngo0cl 37948	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
5		en1eqsn 9285	. . . . . 6 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
6	1	rneqi 5922	. . . . . . . 8 ⊢ ran 𝐺 = ran (1st ‘𝑅)
7		uznzr.2	. . . . . . . 8 ⊢ 𝐻 = (2nd ‘𝑅)
8		uznzr.4	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐻)
9	6, 7, 8	rngo1cl 37968	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10		eleq2 2824	. . . . . . . . . 10 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
11	10	biimpd 229	. . . . . . . . 9 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑈 ∈ {𝑍}))
12		elsni 4623	. . . . . . . . 9 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
13	11, 12	syl6com 37	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
14	2	eqcomi 2745	. . . . . . . 8 ⊢ ran 𝐺 = 𝑋
15	13, 14	eleq2s 2853	. . . . . . 7 ⊢ (𝑈 ∈ ran 𝐺 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
16	9, 15	syl 17	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑈 = 𝑍))
17	5, 16	syl5com 31	. . . . 5 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → (𝑅 ∈ RingOps → 𝑈 = 𝑍))
18	17	ex 412	. . . 4 ⊢ (𝑍 ∈ 𝑋 → (𝑋 ≈ 1o → (𝑅 ∈ RingOps → 𝑈 = 𝑍)))
19	18	com23 86	. . 3 ⊢ (𝑍 ∈ 𝑋 → (𝑅 ∈ RingOps → (𝑋 ≈ 1o → 𝑈 = 𝑍)))
20	4, 19	mpcom 38	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o → 𝑈 = 𝑍))
21	1, 2	rngone0 37940	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22		oveq2 7418	. . . . . 6 ⊢ (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
23	22	ralrimivw 3137	. . . . 5 ⊢ (𝑈 = 𝑍 → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
24	3, 2, 1, 7	rngorz 37952	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑍) = 𝑍)
25	24	ralrimiva 3133	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍)
26	2, 6	eqtri 2759	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
27	7, 26, 8	rngoridm 37967	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑈) = 𝑥)
28	27	ralrimiva 3133	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥)
29		r19.26 3099	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30		r19.26 3099	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍))
31		eqtr 2756	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32		eqtr 2756	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑥𝐻𝑍) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
33	32	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍))
34	31, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍))
35	34	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑥𝐻𝑈) → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍)))
36	35	eqcoms 2744	. . . . . . . . . . . . . . 15 ⊢ ((𝑥𝐻𝑈) = 𝑥 → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍)))
37	36	imp31 417	. . . . . . . . . . . . . 14 ⊢ ((((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
38	37	ralimi 3074	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥 ∈ 𝑋 𝑥 = 𝑍)
39		eqsn 4810	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍))
40		ensn1g 9041	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ≈ 1o)
41	4, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ RingOps → {𝑍} ≈ 1o)
42		breq1 5127	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = {𝑍} → (𝑋 ≈ 1o ↔ {𝑍} ≈ 1o))
43	41, 42	imbitrrid 246	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = {𝑍} → (𝑅 ∈ RingOps → 𝑋 ≈ 1o))
44	39, 43	biimtrrdi 254	. . . . . . . . . . . . . 14 ⊢ (𝑋 ≠ ∅ → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → (𝑅 ∈ RingOps → 𝑋 ≈ 1o)))
45	44	com3l 89	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
46	38, 45	syl 17	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
47	30, 46	sylbir 235	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
48	47	ex 412	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
49	29, 48	sylbir 235	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
50	49	ex 412	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))))
51	50	com24 95	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 → (𝑅 ∈ RingOps → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))))
52	28, 51	mpcom 38	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
53	25, 52	mpd 15	. . . . 5 ⊢ (𝑅 ∈ RingOps → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
54	23, 53	syl5com 31	. . . 4 ⊢ (𝑈 = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
55	54	com13 88	. . 3 ⊢ (𝑋 ≠ ∅ → (𝑅 ∈ RingOps → (𝑈 = 𝑍 → 𝑋 ≈ 1o)))
56	21, 55	mpcom 38	. 2 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 → 𝑋 ≈ 1o))
57	20, 56	impbid 212	1 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∅c0 4313  {csn 4606   class class class wbr 5124  ran crn 5660  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  1_oc1o 8478   ≈ cen 8961  GIdcgi 30476  RingOpscrngo 37923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-1st 7993  df-2nd 7994  df-1o 8485  df-en 8965  df-grpo 30479  df-gid 30480  df-ablo 30531  df-ass 37872  df-exid 37874  df-mgmOLD 37878  df-sgrOLD 37890  df-mndo 37896  df-rngo 37924

This theorem is referenced by:  dvrunz  37983  isdmn3  38103