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Theorem rngoueqz 37112
Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20422 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 = (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 37091 . . 3 (𝑅 ∈ RingOps → 𝑍𝑋)
5 en1eqsn 9278 . . . . . 6 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
61rneqi 5936 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
7 uznzr.2 . . . . . . . 8 𝐻 = (2nd𝑅)
8 uznzr.4 . . . . . . . 8 𝑈 = (GId‘𝐻)
96, 7, 8rngo1cl 37111 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10 eleq2 2821 . . . . . . . . . 10 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
1110biimpd 228 . . . . . . . . 9 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
12 elsni 4645 . . . . . . . . 9 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
1311, 12syl6com 37 . . . . . . . 8 (𝑈𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
142eqcomi 2740 . . . . . . . 8 ran 𝐺 = 𝑋
1513, 14eleq2s 2850 . . . . . . 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 412 . . . 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 37083 . . 3 (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22 oveq2 7420 . . . . . 6 (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
2322ralrimivw 3149 . . . . 5 (𝑈 = 𝑍 → ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
243, 2, 1, 7rngorz 37095 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑍) = 𝑍)
2524ralrimiva 3145 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍)
262, 6eqtri 2759 . . . . . . . . 9 𝑋 = ran (1st𝑅)
277, 26, 8rngoridm 37110 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑈) = 𝑥)
2827ralrimiva 3145 . . . . . . 7 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥)
29 r19.26 3110 . . . . . . . . . 10 (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30 r19.26 3110 . . . . . . . . . . . 12 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍))
31 eqtr 2754 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32 eqtr 2754 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = (𝑥𝐻𝑍) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
3332ex 412 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍))
3534ex 412 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑥𝐻𝑈) → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍)))
3635eqcoms 2739 . . . . . . . . . . . . . . 15 ((𝑥𝐻𝑈) = 𝑥 → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍)))
3736imp31 417 . . . . . . . . . . . . . 14 ((((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
3837ralimi 3082 . . . . . . . . . . . . 13 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥𝑋 𝑥 = 𝑍)
39 eqsn 4832 . . . . . . . . . . . . . . 15 (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥𝑋 𝑥 = 𝑍))
40 ensn1g 9023 . . . . . . . . . . . . . . . . 17 (𝑍𝑋 → {𝑍} ≈ 1o)
414, 40syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → {𝑍} ≈ 1o)
42 breq1 5151 . . . . . . . . . . . . . . . 16 (𝑋 = {𝑍} → (𝑋 ≈ 1o ↔ {𝑍} ≈ 1o))
4341, 42imbitrrid 245 . . . . . . . . . . . . . . 15 (𝑋 = {𝑍} → (𝑅 ∈ RingOps → 𝑋 ≈ 1o))
4439, 43syl6bir 254 . . . . . . . . . . . . . 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 234 . . . . . . . . . . 11 ((∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍) → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o)))
4847ex 412 . . . . . . . . . 10 (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
4929, 48sylbir 234 . . . . . . . . 9 ((∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → (∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍 → (𝑅 ∈ RingOps → (𝑋 ≠ ∅ → 𝑋 ≈ 1o))))
5049ex 412 . . . . . . . 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 211 1 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  c0 4322  {csn 4628   class class class wbr 5148  ran crn 5677  cfv 6543  (class class class)co 7412  1st c1st 7977  2nd c2nd 7978  1oc1o 8463  cen 8940  GIdcgi 30011  RingOpscrngo 37066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-1st 7979  df-2nd 7980  df-1o 8470  df-en 8944  df-grpo 30014  df-gid 30015  df-ablo 30066  df-ass 37015  df-exid 37017  df-mgmOLD 37021  df-sgrOLD 37033  df-mndo 37039  df-rngo 37067
This theorem is referenced by:  dvrunz  37126  isdmn3  37246
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