Theorem rngoueqz 38591

Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20631 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 38570	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
5		en1eqsn 9231	. . . . . 6 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
6	1	rneqi 5927	. . . . . . . 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 38590	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10		eleq2 2852	. . . . . . . . . 10 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
11	10	biimpd 232	. . . . . . . . 9 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑈 ∈ {𝑍}))
12		elsni 4606	. . . . . . . . 9 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
13	11, 12	syl6com 38	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
14	2	eqcomi 2772	. . . . . . . 8 ⊢ ran 𝐺 = 𝑋
15	13, 14	eleq2s 2881	. . . . . . 7 ⊢ (𝑈 ∈ ran 𝐺 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
16	9, 15	syl 18	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑈 = 𝑍))
17	5, 16	syl5com 32	. . . . 5 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → (𝑅 ∈ RingOps → 𝑈 = 𝑍))
18	17	ex 417	. . . 4 ⊢ (𝑍 ∈ 𝑋 → (𝑋 ≈ 1o → (𝑅 ∈ RingOps → 𝑈 = 𝑍)))
19	18	com23 87	. . 3 ⊢ (𝑍 ∈ 𝑋 → (𝑅 ∈ RingOps → (𝑋 ≈ 1o → 𝑈 = 𝑍)))
20	4, 19	mpcom 39	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o → 𝑈 = 𝑍))
21	1, 2	rngone0 38562	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22		oveq2 7418	. . . . . 6 ⊢ (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
23	22	ralrimivw 3161	. . . . 5 ⊢ (𝑈 = 𝑍 → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
24	3, 2, 1, 7	rngorz 38574	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑍) = 𝑍)
25	24	ralrimiva 3157	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍)
26	2, 6	eqtri 2786	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
27	7, 26, 8	rngoridm 38589	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑈) = 𝑥)
28	27	ralrimiva 3157	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥)
29		r19.26 3125	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30		r19.26 3125	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍))
31		eqtr 2783	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32		eqtr 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑥𝐻𝑍) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
33	32	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍))
34	31, 33	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍))
35	34	ex 417	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑥𝐻𝑈) → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍)))
36	35	eqcoms 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑥𝐻𝑈) = 𝑥 → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍 → 𝑥 = 𝑍)))
37	36	imp31 422	. . . . . . . . . . . . . 14 ⊢ ((((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
38	37	ralimi 3102	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥 ∈ 𝑋 𝑥 = 𝑍)
39		eqsn 4795	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍))
40		ensn1g 9015	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ≈ 1o)
41	4, 40	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ RingOps → {𝑍} ≈ 1o)
42		breq1 5112	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = {𝑍} → (𝑋 ≈ 1o ↔ {𝑍} ≈ 1o))
43	41, 42	imbitrrid 249	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = {𝑍} → (𝑅 ∈ RingOps → 𝑋 ≈ 1o))
44	39, 43	biimtrrdi 257	. . . . . . . . . . . . . 14 ⊢ (𝑋 ≠ ∅ → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → (𝑅 ∈ RingOps → 𝑋 ≈ 1o)))
45	44	com3l 90	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
46	38, 45	syl 18	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
47	30, 46	sylbir 238	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
48	47	ex 417	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
49	29, 48	sylbir 238	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
50	49	ex 417	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))))
51	50	com24 96	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 → (𝑅 ∈ RingOps → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))))
52	28, 51	mpcom 39	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍 → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
53	25, 52	mpd 16	. . . . 5 ⊢ (𝑅 ∈ RingOps → (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
54	23, 53	syl5com 32	. . . 4 ⊢ (𝑈 = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
55	54	com13 89	. . 3 ⊢ (𝑋 ≠ ∅ → (𝑅 ∈ RingOps → (𝑈 = 𝑍 → 𝑋 ≈ 1o)))
56	21, 55	mpcom 39	. 2 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 → 𝑋 ≈ 1o))
57	20, 56	impbid 215	1 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∅c0 4286  {csn 4589   class class class wbr 5109  ran crn 5662  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  1_oc1o 8442   ≈ cen 8936  GIdcgi 30842  RingOpscrngo 38545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-suc 6366  df-iota 6492  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 7982  df-2nd 7983  df-1o 8449  df-en 8940  df-grpo 30845  df-gid 30846  df-ablo 30897  df-ass 38494  df-exid 38496  df-mgmOLD 38500  df-sgrOLD 38512  df-mndo 38518  df-rngo 38546

This theorem is referenced by:  dvrunz  38605  isdmn3  38725