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Theorem rngoueqz 33869
Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 19321 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 33848 . . 3 (𝑅 ∈ RingOps → 𝑍𝑋)
5 en1eqsn 8231 . . . . . 6 ((𝑍𝑋𝑋 ≈ 1𝑜) → 𝑋 = {𝑍})
61rneqi 5384 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
7 uznzr.2 . . . . . . . 8 𝐻 = (2nd𝑅)
8 uznzr.4 . . . . . . . 8 𝑈 = (GId‘𝐻)
96, 7, 8rngo1cl 33868 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10 eleq2 2719 . . . . . . . . . 10 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
1110biimpd 219 . . . . . . . . 9 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
12 elsni 4227 . . . . . . . . 9 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
1311, 12syl6com 37 . . . . . . . 8 (𝑈𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
142eqcomi 2660 . . . . . . . 8 ran 𝐺 = 𝑋
1513, 14eleq2s 2748 . . . . . . 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 86 . . 3 (𝑍𝑋 → (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍)))
204, 19mpcom 38 . 2 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))
211, 2rngone0 33840 . . 3 (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22 oveq2 6698 . . . . . 6 (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
2322ralrimivw 2996 . . . . 5 (𝑈 = 𝑍 → ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
243, 2, 1, 7rngorz 33852 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑍) = 𝑍)
2524ralrimiva 2995 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍)
262, 6eqtri 2673 . . . . . . . . 9 𝑋 = ran (1st𝑅)
277, 26, 8rngoridm 33867 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑈) = 𝑥)
2827ralrimiva 2995 . . . . . . 7 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥)
29 r19.26 3093 . . . . . . . . . 10 (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30 r19.26 3093 . . . . . . . . . . . 12 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍))
31 eqtr 2670 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32 eqtr 2670 . . . . . . . . . . . . . . . . . . 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 2659 . . . . . . . . . . . . . . 15 ((𝑥𝐻𝑈) = 𝑥 → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍)))
3736imp31 447 . . . . . . . . . . . . . 14 ((((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
3837ralimi 2981 . . . . . . . . . . . . 13 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥𝑋 𝑥 = 𝑍)
39 eqsn 4393 . . . . . . . . . . . . . . 15 (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥𝑋 𝑥 = 𝑍))
40 ensn1g 8062 . . . . . . . . . . . . . . . . 17 (𝑍𝑋 → {𝑍} ≈ 1𝑜)
414, 40syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → {𝑍} ≈ 1𝑜)
42 breq1 4688 . . . . . . . . . . . . . . . 16 (𝑋 = {𝑍} → (𝑋 ≈ 1𝑜 ↔ {𝑍} ≈ 1𝑜))
4341, 42syl5ibr 236 . . . . . . . . . . . . . . 15 (𝑋 = {𝑍} → (𝑅 ∈ RingOps → 𝑋 ≈ 1𝑜))
4439, 43syl6bir 244 . . . . . . . . . . . . . 14 (𝑋 ≠ ∅ → (∀𝑥𝑋 𝑥 = 𝑍 → (𝑅 ∈ RingOps → 𝑋 ≈ 1𝑜)))
4544com3l 89 . . . . . . . . . . . . 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 225 . . . . . . . . . . 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 225 . . . . . . . . 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 95 . . . . . . 7 (∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 → (𝑅 ∈ RingOps → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍) → (𝑋 ≠ ∅ → 𝑋 ≈ 1𝑜)))))
5228, 51mpcom 38 . . . . . 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 88 . . 3 (𝑋 ≠ ∅ → (𝑅 ∈ RingOps → (𝑈 = 𝑍𝑋 ≈ 1𝑜)))
5621, 55mpcom 38 . 2 (𝑅 ∈ RingOps → (𝑈 = 𝑍𝑋 ≈ 1𝑜))
5720, 56impbid 202 1 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  c0 3948  {csn 4210   class class class wbr 4685  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  1𝑜c1o 7598  cen 7994  GIdcgi 27472  RingOpscrngo 33823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-grpo 27475  df-gid 27476  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824
This theorem is referenced by:  dvrunz  33883  isdmn3  34003
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