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Theorem rngoidmlem 36445
Description: The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
uridm.1 𝐻 = (2nd𝑅)
uridm.2 𝑋 = ran (1st𝑅)
uridm.3 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
rngoidmlem ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

Proof of Theorem rngoidmlem
StepHypRef Expression
1 uridm.1 . . . . 5 𝐻 = (2nd𝑅)
21rngomndo 36444 . . . 4 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3 mndomgmid 36380 . . . 4 (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId ))
4 eqid 2733 . . . . . 6 ran 𝐻 = ran 𝐻
5 uridm.3 . . . . . 6 𝑈 = (GId‘𝐻)
64, 5cmpidelt 36368 . . . . 5 ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
76ex 414 . . . 4 (𝐻 ∈ (Magma ∩ ExId ) → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
82, 3, 73syl 18 . . 3 (𝑅 ∈ RingOps → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
9 eqid 2733 . . . . 5 (1st𝑅) = (1st𝑅)
101, 9rngorn1eq 36443 . . . 4 (𝑅 ∈ RingOps → ran (1st𝑅) = ran 𝐻)
11 uridm.2 . . . . 5 𝑋 = ran (1st𝑅)
12 eqtr 2756 . . . . . 6 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran 𝐻) → 𝑋 = ran 𝐻)
13 simpl 484 . . . . . . . . 9 ((𝑋 = ran 𝐻𝑅 ∈ RingOps) → 𝑋 = ran 𝐻)
1413eleq2d 2820 . . . . . . . 8 ((𝑋 = ran 𝐻𝑅 ∈ RingOps) → (𝐴𝑋𝐴 ∈ ran 𝐻))
1514imbi1d 342 . . . . . . 7 ((𝑋 = ran 𝐻𝑅 ∈ RingOps) → ((𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
1615ex 414 . . . . . 6 (𝑋 = ran 𝐻 → (𝑅 ∈ RingOps → ((𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))))
1712, 16syl 17 . . . . 5 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran 𝐻) → (𝑅 ∈ RingOps → ((𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))))
1811, 17mpan 689 . . . 4 (ran (1st𝑅) = ran 𝐻 → (𝑅 ∈ RingOps → ((𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))))
1910, 18mpcom 38 . . 3 (𝑅 ∈ RingOps → ((𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
208, 19mpbird 257 . 2 (𝑅 ∈ RingOps → (𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
2120imp 408 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cin 3913  ran crn 5638  cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  GIdcgi 29481   ExId cexid 36353  Magmacmagm 36357  MndOpcmndo 36375  RingOpscrngo 36403
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 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-riota 7317  df-ov 7364  df-1st 7925  df-2nd 7926  df-grpo 29484  df-gid 29485  df-ablo 29536  df-ass 36352  df-exid 36354  df-mgmOLD 36358  df-sgrOLD 36370  df-mndo 36376  df-rngo 36404
This theorem is referenced by:  rngolidm  36446  rngoridm  36447
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