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Theorem rngomndo 37107
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 = (2nd β€˜π‘…)
Assertion
Ref Expression
rngomndo (𝑅 ∈ RingOps β†’ 𝐻 ∈ MndOp)

Proof of Theorem rngomndo
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . . 4 (1st β€˜π‘…) = (1st β€˜π‘…)
2 unmnd.1 . . . 4 𝐻 = (2nd β€˜π‘…)
3 eqid 2731 . . . 4 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3rngosm 37072 . . 3 (𝑅 ∈ RingOps β†’ 𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…))
51, 2, 3rngoass 37078 . . . 4 ((𝑅 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…) ∧ 𝑧 ∈ ran (1st β€˜π‘…))) β†’ ((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)))
65ralrimivvva 3202 . . 3 (𝑅 ∈ RingOps β†’ βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)βˆ€π‘§ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)))
71, 2, 3rngoi 37071 . . . 4 (𝑅 ∈ RingOps β†’ (((1st β€˜π‘…) ∈ AbelOp ∧ 𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…)) ∧ (βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)βˆ€π‘§ ∈ ran (1st β€˜π‘…)(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦(1st β€˜π‘…)𝑧)) = ((π‘₯𝐻𝑦)(1st β€˜π‘…)(π‘₯𝐻𝑧)) ∧ ((π‘₯(1st β€˜π‘…)𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)(1st β€˜π‘…)(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
87simprrd 771 . . 3 (𝑅 ∈ RingOps β†’ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
92, 1rngorn1 37105 . . . 4 (𝑅 ∈ RingOps β†’ ran (1st β€˜π‘…) = dom dom 𝐻)
10 xpid11 5931 . . . . . . . 8 ((dom dom 𝐻 Γ— dom dom 𝐻) = (ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…)) ↔ dom dom 𝐻 = ran (1st β€˜π‘…))
1110biimpri 227 . . . . . . 7 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ (dom dom 𝐻 Γ— dom dom 𝐻) = (ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…)))
12 feq23 6701 . . . . . . 7 (((dom dom 𝐻 Γ— dom dom 𝐻) = (ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…)) ∧ dom dom 𝐻 = ran (1st β€˜π‘…)) β†’ (𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ↔ 𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…)))
1311, 12mpancom 685 . . . . . 6 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ (𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ↔ 𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…)))
14 raleq 3321 . . . . . . . 8 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ (βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ↔ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧))))
1514raleqbi1dv 3332 . . . . . . 7 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ (βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ↔ βˆ€π‘¦ ∈ ran (1st β€˜π‘…)βˆ€π‘§ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧))))
1615raleqbi1dv 3332 . . . . . 6 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ (βˆ€π‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ↔ βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)βˆ€π‘§ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧))))
17 raleq 3321 . . . . . . 7 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ (βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦) ↔ βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
1817rexeqbi1dv 3333 . . . . . 6 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ (βˆƒπ‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦) ↔ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
1913, 16, 183anbi123d 1435 . . . . 5 (dom dom 𝐻 = ran (1st β€˜π‘…) β†’ ((𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ∧ βˆ€π‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)) ↔ (𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…) ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)βˆ€π‘§ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
2019eqcoms 2739 . . . 4 (ran (1st β€˜π‘…) = dom dom 𝐻 β†’ ((𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ∧ βˆ€π‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)) ↔ (𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…) ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)βˆ€π‘§ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
219, 20syl 17 . . 3 (𝑅 ∈ RingOps β†’ ((𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ∧ βˆ€π‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)) ↔ (𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…) ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)βˆ€π‘§ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
224, 6, 8, 21mpbir3and 1341 . 2 (𝑅 ∈ RingOps β†’ (𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ∧ βˆ€π‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
23 fvex 6904 . . . 4 (2nd β€˜π‘…) ∈ V
24 eleq1 2820 . . . 4 (𝐻 = (2nd β€˜π‘…) β†’ (𝐻 ∈ V ↔ (2nd β€˜π‘…) ∈ V))
2523, 24mpbiri 258 . . 3 (𝐻 = (2nd β€˜π‘…) β†’ 𝐻 ∈ V)
26 eqid 2731 . . . 4 dom dom 𝐻 = dom dom 𝐻
2726ismndo1 37045 . . 3 (𝐻 ∈ V β†’ (𝐻 ∈ MndOp ↔ (𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ∧ βˆ€π‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
282, 25, 27mp2b 10 . 2 (𝐻 ∈ MndOp ↔ (𝐻:(dom dom 𝐻 Γ— dom dom 𝐻)⟢dom dom 𝐻 ∧ βˆ€π‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom π»βˆ€π‘§ ∈ dom dom 𝐻((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ βˆƒπ‘₯ ∈ dom dom π»βˆ€π‘¦ ∈ dom dom 𝐻((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
2922, 28sylibr 233 1 (𝑅 ∈ RingOps β†’ 𝐻 ∈ MndOp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  Vcvv 3473   Γ— cxp 5674  dom cdm 5676  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  AbelOpcablo 30065  MndOpcmndo 37038  RingOpscrngo 37066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7415  df-1st 7978  df-2nd 7979  df-grpo 30014  df-ablo 30066  df-ass 37015  df-exid 37017  df-mgmOLD 37021  df-sgrOLD 37033  df-mndo 37039  df-rngo 37067
This theorem is referenced by:  rngoidmlem  37108  rngo1cl  37111  isdrngo2  37130
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