Theorem rngoueqz 36472

Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20217 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 36451	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
5		en1eqsn 9225	. . . . . 6 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
6	1	rneqi 5897	. . . . . . . 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 36471	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10		eleq2 2821	. . . . . . . . . 10 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
11	10	biimpd 228	. . . . . . . . 9 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑈 ∈ {𝑍}))
12		elsni 4608	. . . . . . . . 9 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
13	11, 12	syl6com 37	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
14	2	eqcomi 2740	. . . . . . . 8 ⊢ ran 𝐺 = 𝑋
15	13, 14	eleq2s 2850	. . . . . . 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 413	. . . 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 36443	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22		oveq2 7370	. . . . . 6 ⊢ (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
23	22	ralrimivw 3143	. . . . 5 ⊢ (𝑈 = 𝑍 → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
24	3, 2, 1, 7	rngorz 36455	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑍) = 𝑍)
25	24	ralrimiva 3139	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍)
26	2, 6	eqtri 2759	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
27	7, 26, 8	rngoridm 36470	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑈) = 𝑥)
28	27	ralrimiva 3139	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥)
29		r19.26 3110	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30		r19.26 3110	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍))
31		eqtr 2754	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32		eqtr 2754	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑥𝐻𝑍) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
33	32	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍))
34	31, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍))
35	34	ex 413	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑥𝐻𝑈) → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍)))
36	35	eqcoms 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝑥𝐻𝑈) = 𝑥 → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍)))
37	36	imp31 418	. . . . . . . . . . . . . 14 ⊢ ((((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
38	37	ralimi 3082	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥 ∈ 𝑋 𝑥 = 𝑍)
39		eqsn 4794	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍))
40		ensn1g 8970	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ≈ 1o)
41	4, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ RingOps → {𝑍} ≈ 1o)
42		breq1 5113	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = {𝑍} → (𝑋 ≈ 1o ↔ {𝑍} ≈ 1o))
43	41, 42	imbitrrid 245	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = {𝑍} → (𝑅 ∈ RingOps → 𝑋 ≈ 1o))
44	39, 43	syl6bir 253	. . . . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
48	47	ex 413	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
49	29, 48	sylbir 234	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
50	49	ex 413	. . . . . . . 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 211	1 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∅c0 4287  {csn 4591   class class class wbr 5110  ran crn 5639  ‘cfv 6501  (class class class)co 7362  1^st c1st 7924  2^nd c2nd 7925  1_oc1o 8410   ≈ cen 8887  GIdcgi 29495  RingOpscrngo 36426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-1st 7926  df-2nd 7927  df-1o 8417  df-en 8891  df-grpo 29498  df-gid 29499  df-ablo 29550  df-ass 36375  df-exid 36377  df-mgmOLD 36381  df-sgrOLD 36393  df-mndo 36399  df-rngo 36427

This theorem is referenced by:  dvrunz  36486  isdmn3  36606