Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngoidmlem Structured version   Visualization version   GIF version

Theorem rngoidmlem 36395
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 36394 . . . 4 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3 mndomgmid 36330 . . . 4 (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId ))
4 eqid 2736 . . . . . 6 ran 𝐻 = ran 𝐻
5 uridm.3 . . . . . 6 𝑈 = (GId‘𝐻)
64, 5cmpidelt 36318 . . . . 5 ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
76ex 413 . . . 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 2736 . . . . 5 (1st𝑅) = (1st𝑅)
101, 9rngorn1eq 36393 . . . 4 (𝑅 ∈ RingOps → ran (1st𝑅) = ran 𝐻)
11 uridm.2 . . . . 5 𝑋 = ran (1st𝑅)
12 eqtr 2759 . . . . . 6 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran 𝐻) → 𝑋 = ran 𝐻)
13 simpl 483 . . . . . . . . 9 ((𝑋 = ran 𝐻𝑅 ∈ RingOps) → 𝑋 = ran 𝐻)
1413eleq2d 2823 . . . . . . . 8 ((𝑋 = ran 𝐻𝑅 ∈ RingOps) → (𝐴𝑋𝐴 ∈ ran 𝐻))
1514imbi1d 341 . . . . . . 7 ((𝑋 = ran 𝐻𝑅 ∈ RingOps) → ((𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
1615ex 413 . . . . . 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 688 . . . 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 256 . 2 (𝑅 ∈ RingOps → (𝐴𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
2120imp 407 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  cin 3909  ran crn 5634  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  GIdcgi 29432   ExId cexid 36303  Magmacmagm 36307  MndOpcmndo 36325  RingOpscrngo 36353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-riota 7313  df-ov 7360  df-1st 7921  df-2nd 7922  df-grpo 29435  df-gid 29436  df-ablo 29487  df-ass 36302  df-exid 36304  df-mgmOLD 36308  df-sgrOLD 36320  df-mndo 36326  df-rngo 36354
This theorem is referenced by:  rngolidm  36396  rngoridm  36397
  Copyright terms: Public domain W3C validator