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Theorem rngoueqz 37948
Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20533 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 37927 . . 3 (𝑅 ∈ RingOps → 𝑍𝑋)
5 en1eqsn 9309 . . . . . 6 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
61rneqi 5947 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
7 uznzr.2 . . . . . . . 8 𝐻 = (2nd𝑅)
8 uznzr.4 . . . . . . . 8 𝑈 = (GId‘𝐻)
96, 7, 8rngo1cl 37947 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10 eleq2 2829 . . . . . . . . . 10 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
1110biimpd 229 . . . . . . . . 9 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
12 elsni 4642 . . . . . . . . 9 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
1311, 12syl6com 37 . . . . . . . 8 (𝑈𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
142eqcomi 2745 . . . . . . . 8 ran 𝐺 = 𝑋
1513, 14eleq2s 2858 . . . . . . 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 37919 . . 3 (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22 oveq2 7440 . . . . . 6 (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
2322ralrimivw 3149 . . . . 5 (𝑈 = 𝑍 → ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
243, 2, 1, 7rngorz 37931 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑍) = 𝑍)
2524ralrimiva 3145 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍)
262, 6eqtri 2764 . . . . . . . . 9 𝑋 = ran (1st𝑅)
277, 26, 8rngoridm 37946 . . . . . . . 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 2759 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32 eqtr 2759 . . . . . . . . . . . . . . . . . . 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 2744 . . . . . . . . . . . . . . 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 4828 . . . . . . . . . . . . . . 15 (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥𝑋 𝑥 = 𝑍))
40 ensn1g 9063 . . . . . . . . . . . . . . . . 17 (𝑍𝑋 → {𝑍} ≈ 1o)
414, 40syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → {𝑍} ≈ 1o)
42 breq1 5145 . . . . . . . . . . . . . . . 16 (𝑋 = {𝑍} → (𝑋 ≈ 1o ↔ {𝑍} ≈ 1o))
4341, 42imbitrrid 246 . . . . . . . . . . . . . . 15 (𝑋 = {𝑍} → (𝑅 ∈ RingOps → 𝑋 ≈ 1o))
4439, 43biimtrrdi 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 235 . . . . . . . . . . 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 235 . . . . . . . . 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 212 1 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  c0 4332  {csn 4625   class class class wbr 5142  ran crn 5685  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  1oc1o 8500  cen 8983  GIdcgi 30510  RingOpscrngo 37902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-1st 8015  df-2nd 8016  df-1o 8507  df-en 8987  df-grpo 30513  df-gid 30514  df-ablo 30565  df-ass 37851  df-exid 37853  df-mgmOLD 37857  df-sgrOLD 37869  df-mndo 37875  df-rngo 37903
This theorem is referenced by:  dvrunz  37962  isdmn3  38082
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