Theorem rngoueqz 36399

Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20743 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 36378	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
5		en1eqsn 9218	. . . . . 6 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
6	1	rneqi 5892	. . . . . . . 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 36398	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10		eleq2 2826	. . . . . . . . . 10 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
11	10	biimpd 228	. . . . . . . . 9 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑈 ∈ {𝑍}))
12		elsni 4603	. . . . . . . . 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 2856	. . . . . . 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 36370	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22		oveq2 7365	. . . . . 6 ⊢ (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
23	22	ralrimivw 3147	. . . . 5 ⊢ (𝑈 = 𝑍 → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
24	3, 2, 1, 7	rngorz 36382	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑍) = 𝑍)
25	24	ralrimiva 3143	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍)
26	2, 6	eqtri 2764	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
27	7, 26, 8	rngoridm 36397	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑥𝐻𝑈) = 𝑥)
28	27	ralrimiva 3143	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥)
29		r19.26 3114	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30		r19.26 3114	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥 ∈ 𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥 ∈ 𝑋 (𝑥𝐻𝑍) = 𝑍))
31		eqtr 2759	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32		eqtr 2759	. . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . 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 3086	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥 ∈ 𝑋 𝑥 = 𝑍)
39		eqsn 4789	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍))
40		ensn1g 8963	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ≈ 1o)
41	4, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ RingOps → {𝑍} ≈ 1o)
42		breq1 5108	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = {𝑍} → (𝑋 ≈ 1o ↔ {𝑍} ≈ 1o))
43	41, 42	syl5ibr 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 2943  ∀wral 3064  ∅c0 4282  {csn 4586   class class class wbr 5105  ran crn 5634  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920  1_oc1o 8405   ≈ cen 8880  GIdcgi 29432  RingOpscrngo 36353

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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-1st 7921  df-2nd 7922  df-1o 8412  df-en 8884  df-grpo 29435  df-gid 29436  df-ablo 29487  df-ass 36302  df-exid 36304  df-mgmOLD 36308  df-sgrOLD 36320  df-mndo 36326  df-rngo 36354

This theorem is referenced by:  dvrunz  36413  isdmn3  36533