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

Theorem rngomndo 37844
Description: In a unital ring the multiplication is a monoid. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
unmnd.1 𝐻 = (2nd𝑅)
Assertion
Ref Expression
rngomndo (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Proof of Theorem rngomndo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . 4 (1st𝑅) = (1st𝑅)
2 unmnd.1 . . . 4 𝐻 = (2nd𝑅)
3 eqid 2734 . . . 4 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3rngosm 37809 . . 3 (𝑅 ∈ RingOps → 𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅))
51, 2, 3rngoass 37815 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
65ralrimivvva 3207 . . 3 (𝑅 ∈ RingOps → ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
71, 2, 3rngoi 37808 . . . 4 (𝑅 ∈ RingOps → (((1st𝑅) ∈ AbelOp ∧ 𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅)) ∧ (∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦(1st𝑅)𝑧)) = ((𝑥𝐻𝑦)(1st𝑅)(𝑥𝐻𝑧)) ∧ ((𝑥(1st𝑅)𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)(1st𝑅)(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
87simprrd 773 . . 3 (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
92, 1rngorn1 37842 . . . 4 (𝑅 ∈ RingOps → ran (1st𝑅) = dom dom 𝐻)
10 xpid11 5956 . . . . . . . 8 ((dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)) ↔ dom dom 𝐻 = ran (1st𝑅))
1110biimpri 228 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)))
12 feq23 6730 . . . . . . 7 (((dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)) ∧ dom dom 𝐻 = ran (1st𝑅)) → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅)))
1311, 12mpancom 687 . . . . . 6 (dom dom 𝐻 = ran (1st𝑅) → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅)))
14 raleq 3326 . . . . . . . 8 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
1514raleqbi1dv 3341 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
1615raleqbi1dv 3341 . . . . . 6 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
17 raleq 3326 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
1817rexeqbi1dv 3342 . . . . . 6 (dom dom 𝐻 = ran (1st𝑅) → (∃𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
1913, 16, 183anbi123d 1436 . . . . 5 (dom dom 𝐻 = ran (1st𝑅) → ((𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) ↔ (𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅) ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
2019eqcoms 2742 . . . 4 (ran (1st𝑅) = dom dom 𝐻 → ((𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) ↔ (𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅) ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
219, 20syl 17 . . 3 (𝑅 ∈ RingOps → ((𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) ↔ (𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅) ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
224, 6, 8, 21mpbir3and 1342 . 2 (𝑅 ∈ RingOps → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
23 fvex 6932 . . . 4 (2nd𝑅) ∈ V
24 eleq1 2826 . . . 4 (𝐻 = (2nd𝑅) → (𝐻 ∈ V ↔ (2nd𝑅) ∈ V))
2523, 24mpbiri 258 . . 3 (𝐻 = (2nd𝑅) → 𝐻 ∈ V)
26 eqid 2734 . . . 4 dom dom 𝐻 = dom dom 𝐻
2726ismndo1 37782 . . 3 (𝐻 ∈ V → (𝐻 ∈ MndOp ↔ (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
282, 25, 27mp2b 10 . 2 (𝐻 ∈ MndOp ↔ (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
2922, 28sylibr 234 1 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wral 3063  wrex 3072  Vcvv 3482   × cxp 5697  dom cdm 5699  ran crn 5700  wf 6568  cfv 6572  (class class class)co 7445  1st c1st 8024  2nd c2nd 8025  AbelOpcablo 30567  MndOpcmndo 37775  RingOpscrngo 37803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fo 6578  df-fv 6580  df-ov 7448  df-1st 8026  df-2nd 8027  df-grpo 30516  df-ablo 30568  df-ass 37752  df-exid 37754  df-mgmOLD 37758  df-sgrOLD 37770  df-mndo 37776  df-rngo 37804
This theorem is referenced by:  rngoidmlem  37845  rngo1cl  37848  isdrngo2  37867
  Copyright terms: Public domain W3C validator