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Theorem rngomndo 37441
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 2728 . . . 4 (1st𝑅) = (1st𝑅)
2 unmnd.1 . . . 4 𝐻 = (2nd𝑅)
3 eqid 2728 . . . 4 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3rngosm 37406 . . 3 (𝑅 ∈ RingOps → 𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅))
51, 2, 3rngoass 37412 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
65ralrimivvva 3201 . . 3 (𝑅 ∈ RingOps → ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
71, 2, 3rngoi 37405 . . . 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 772 . . 3 (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
92, 1rngorn1 37439 . . . 4 (𝑅 ∈ RingOps → ran (1st𝑅) = dom dom 𝐻)
10 xpid11 5938 . . . . . . . 8 ((dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)) ↔ dom dom 𝐻 = ran (1st𝑅))
1110biimpri 227 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)))
12 feq23 6711 . . . . . . 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 686 . . . . . 6 (dom dom 𝐻 = ran (1st𝑅) → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅)))
14 raleq 3320 . . . . . . . 8 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
1514raleqbi1dv 3331 . . . . . . 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 3331 . . . . . 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 3320 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
1817rexeqbi1dv 3332 . . . . . 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 1432 . . . . 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 2736 . . . 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 1339 . 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 6915 . . . 4 (2nd𝑅) ∈ V
24 eleq1 2817 . . . 4 (𝐻 = (2nd𝑅) → (𝐻 ∈ V ↔ (2nd𝑅) ∈ V))
2523, 24mpbiri 257 . . 3 (𝐻 = (2nd𝑅) → 𝐻 ∈ V)
26 eqid 2728 . . . 4 dom dom 𝐻 = dom dom 𝐻
2726ismndo1 37379 . . 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 233 1 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wrex 3067  Vcvv 3473   × cxp 5680  dom cdm 5682  ran crn 5683  wf 6549  cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  AbelOpcablo 30374  MndOpcmndo 37372  RingOpscrngo 37400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-ov 7429  df-1st 7999  df-2nd 8000  df-grpo 30323  df-ablo 30375  df-ass 37349  df-exid 37351  df-mgmOLD 37355  df-sgrOLD 37367  df-mndo 37373  df-rngo 37401
This theorem is referenced by:  rngoidmlem  37442  rngo1cl  37445  isdrngo2  37464
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