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Theorem rngoueqz 32909
 Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 19094 instead. In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
uznzr.1 𝐺 = (1st𝑅)
uznzr.2 𝐻 = (2nd𝑅)
uznzr.3 𝑍 = (GId‘𝐺)
uznzr.4 𝑈 = (GId‘𝐻)
uznzr.5 𝑋 = ran 𝐺
Assertion
Ref Expression
rngoueqz (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))

Proof of Theorem rngoueqz
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 uznzr.1 . . . 4 𝐺 = (1st𝑅)
2 uznzr.5 . . . 4 𝑋 = ran 𝐺
3 uznzr.3 . . . 4 𝑍 = (GId‘𝐺)
41, 2, 3rngo0cl 32888 . . 3 (𝑅 ∈ RingOps → 𝑍𝑋)
5 en1eqsn 8075 . . . . . 6 ((𝑍𝑋𝑋 ≈ 1𝑜) → 𝑋 = {𝑍})
61rneqi 5273 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
7 uznzr.2 . . . . . . . 8 𝐻 = (2nd𝑅)
8 uznzr.4 . . . . . . . 8 𝑈 = (GId‘𝐻)
96, 7, 8rngo1cl 32908 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10 eleq2 2677 . . . . . . . . . 10 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
1110biimpd 218 . . . . . . . . 9 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
12 elsni 4142 . . . . . . . . 9 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
1311, 12syl6com 36 . . . . . . . 8 (𝑈𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
142eqcomi 2619 . . . . . . . 8 ran 𝐺 = 𝑋
1513, 14eleq2s 2706 . . . . . . 7 (𝑈 ∈ ran 𝐺 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
169, 15syl 17 . . . . . 6 (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑈 = 𝑍))
175, 16syl5com 31 . . . . 5 ((𝑍𝑋𝑋 ≈ 1𝑜) → (𝑅 ∈ RingOps → 𝑈 = 𝑍))
1817ex 449 . . . 4 (𝑍𝑋 → (𝑋 ≈ 1𝑜 → (𝑅 ∈ RingOps → 𝑈 = 𝑍)))
1918com23 84 . . 3 (𝑍𝑋 → (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍)))
204, 19mpcom 37 . 2 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))
211, 2rngone0 32880 . . 3 (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22 oveq2 6557 . . . . . 6 (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
2322ralrimivw 2950 . . . . 5 (𝑈 = 𝑍 → ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
243, 2, 1, 7rngorz 32892 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑍) = 𝑍)
2524ralrimiva 2949 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍)
262, 6eqtri 2632 . . . . . . . . 9 𝑋 = ran (1st𝑅)
277, 26, 8rngoridm 32907 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑈) = 𝑥)
2827ralrimiva 2949 . . . . . . 7 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥)
29 r19.26 3046 . . . . . . . . . 10 (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30 r19.26 3046 . . . . . . . . . . . 12 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍))
31 eqtr 2629 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32 eqtr 2629 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = (𝑥𝐻𝑍) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
3332ex 449 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍))
3534ex 449 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑥𝐻𝑈) → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍)))
3635eqcoms 2618 . . . . . . . . . . . . . . 15 ((𝑥𝐻𝑈) = 𝑥 → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍)))
3736imp31 447 . . . . . . . . . . . . . 14 ((((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
3837ralimi 2936 . . . . . . . . . . . . 13 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥𝑋 𝑥 = 𝑍)
39 eqsn 4301 . . . . . . . . . . . . . . 15 (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥𝑋 𝑥 = 𝑍))
40 ensn1g 7907 . . . . . . . . . . . . . . . . 17 (𝑍𝑋 → {𝑍} ≈ 1𝑜)
414, 40syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → {𝑍} ≈ 1𝑜)
42 breq1 4586 . . . . . . . . . . . . . . . 16 (𝑋 = {𝑍} → (𝑋 ≈ 1𝑜 ↔ {𝑍} ≈ 1𝑜))
4341, 42syl5ibr 235 . . . . . . . . . . . . . . 15 (𝑋 = {𝑍} → (𝑅 ∈ RingOps → 𝑋 ≈ 1𝑜))
4439, 43syl6bir 243 . . . . . . . . . . . . . 14 (𝑋 ≠ ∅ → (∀𝑥𝑋 𝑥 = 𝑍 → (𝑅 ∈ RingOps → 𝑋 ≈ 1𝑜)))
4544com3l 87 . . . . . . . . . . . . 13 (∀𝑥𝑋 𝑥 = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))
4638, 45syl 17 . . . . . . . . . . . 12 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))
4730, 46sylbir 224 . . . . . . . . . . 11 ((∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))
4847ex 449 . . . . . . . . . 10 (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜))))
4929, 48sylbir 224 . . . . . . . . 9 ((∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜))))
5049ex 449 . . . . . . . 8 (∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 → (∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))))
5150com24 93 . . . . . . 7 (∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 → (𝑅 ∈ RingOps → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))))
5228, 51mpcom 37 . . . . . 6 (𝑅 ∈ RingOps → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜))))
5325, 52mpd 15 . . . . 5 (𝑅 ∈ RingOps → (∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))
5423, 53syl5com 31 . . . 4 (𝑈 = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))
5554com13 86 . . 3 (𝑋 ≠ ∅ → (𝑅 ∈ RingOps → (𝑈 = 𝑍𝑋 ≈ 1𝑜)))
5621, 55mpcom 37 . 2 (𝑅 ∈ RingOps → (𝑈 = 𝑍𝑋 ≈ 1𝑜))
5720, 56impbid 201 1 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874  {csn 4125   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  1𝑜c1o 7440   ≈ cen 7838  GIdcgi 26728  RingOpscrngo 32863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-grpo 26731  df-gid 26732  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864 This theorem is referenced by:  dvrunz  32923  isdmn3  33043
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