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Theorem rngoueqz 36428
Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20759 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 36407 . . 3 (𝑅 ∈ RingOps → 𝑍𝑋)
5 en1eqsn 9225 . . . . . 6 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
61rneqi 5897 . . . . . . . 8 ran 𝐺 = ran (1st𝑅)
7 uznzr.2 . . . . . . . 8 𝐻 = (2nd𝑅)
8 uznzr.4 . . . . . . . 8 𝑈 = (GId‘𝐻)
96, 7, 8rngo1cl 36427 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈 ∈ ran 𝐺)
10 eleq2 2827 . . . . . . . . . 10 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
1110biimpd 228 . . . . . . . . 9 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
12 elsni 4608 . . . . . . . . 9 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
1311, 12syl6com 37 . . . . . . . 8 (𝑈𝑋 → (𝑋 = {𝑍} → 𝑈 = 𝑍))
142eqcomi 2746 . . . . . . . 8 ran 𝐺 = 𝑋
1513, 14eleq2s 2856 . . . . . . 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 414 . . . 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 36399 . . 3 (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
22 oveq2 7370 . . . . . 6 (𝑈 = 𝑍 → (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
2322ralrimivw 3148 . . . . 5 (𝑈 = 𝑍 → ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍))
243, 2, 1, 7rngorz 36411 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑍) = 𝑍)
2524ralrimiva 3144 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍)
262, 6eqtri 2765 . . . . . . . . 9 𝑋 = ran (1st𝑅)
277, 26, 8rngoridm 36426 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑥𝐻𝑈) = 𝑥)
2827ralrimiva 3144 . . . . . . 7 (𝑅 ∈ RingOps → ∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥)
29 r19.26 3115 . . . . . . . . . 10 (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ↔ (∀𝑥𝑋 (𝑥𝐻𝑈) = 𝑥 ∧ ∀𝑥𝑋 (𝑥𝐻𝑈) = (𝑥𝐻𝑍)))
30 r19.26 3115 . . . . . . . . . . . 12 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) ↔ (∀𝑥𝑋 ((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ ∀𝑥𝑋 (𝑥𝐻𝑍) = 𝑍))
31 eqtr 2760 . . . . . . . . . . . . . . . . . 18 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → 𝑥 = (𝑥𝐻𝑍))
32 eqtr 2760 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = (𝑥𝐻𝑍) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
3332ex 414 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 ((𝑥 = (𝑥𝐻𝑈) ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍))
3534ex 414 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑥𝐻𝑈) → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍)))
3635eqcoms 2745 . . . . . . . . . . . . . . 15 ((𝑥𝐻𝑈) = 𝑥 → ((𝑥𝐻𝑈) = (𝑥𝐻𝑍) → ((𝑥𝐻𝑍) = 𝑍𝑥 = 𝑍)))
3736imp31 419 . . . . . . . . . . . . . 14 ((((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → 𝑥 = 𝑍)
3837ralimi 3087 . . . . . . . . . . . . 13 (∀𝑥𝑋 (((𝑥𝐻𝑈) = 𝑥 ∧ (𝑥𝐻𝑈) = (𝑥𝐻𝑍)) ∧ (𝑥𝐻𝑍) = 𝑍) → ∀𝑥𝑋 𝑥 = 𝑍)
39 eqsn 4794 . . . . . . . . . . . . . . 15 (𝑋 ≠ ∅ → (𝑋 = {𝑍} ↔ ∀𝑥𝑋 𝑥 = 𝑍))
40 ensn1g 8970 . . . . . . . . . . . . . . . . 17 (𝑍𝑋 → {𝑍} ≈ 1o)
414, 40syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → {𝑍} ≈ 1o)
42 breq1 5113 . . . . . . . . . . . . . . . 16 (𝑋 = {𝑍} → (𝑋 ≈ 1o ↔ {𝑍} ≈ 1o))
4341, 42syl5ibr 246 . . . . . . . . . . . . . . 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 414 . . . . . . . . . 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 414 . . . . . . . 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 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  c0 4287  {csn 4591   class class class wbr 5110  ran crn 5639  cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  1oc1o 8410  cen 8887  GIdcgi 29474  RingOpscrngo 36382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-1st 7926  df-2nd 7927  df-1o 8417  df-en 8891  df-grpo 29477  df-gid 29478  df-ablo 29529  df-ass 36331  df-exid 36333  df-mgmOLD 36337  df-sgrOLD 36349  df-mndo 36355  df-rngo 36383
This theorem is referenced by:  dvrunz  36442  isdmn3  36562
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