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Theorem rngomndo 37106
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 2730 . . . 4 (1st β€˜π‘…) = (1st β€˜π‘…)
2 unmnd.1 . . . 4 𝐻 = (2nd β€˜π‘…)
3 eqid 2730 . . . 4 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3rngosm 37071 . . 3 (𝑅 ∈ RingOps β†’ 𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…))
51, 2, 3rngoass 37077 . . . 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 3201 . . 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 37070 . . . 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 770 . . 3 (𝑅 ∈ RingOps β†’ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
92, 1rngorn1 37104 . . . 4 (𝑅 ∈ RingOps β†’ ran (1st β€˜π‘…) = dom dom 𝐻)
10 xpid11 5930 . . . . . . . 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 6700 . . . . . . 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 684 . . . . . 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 3320 . . . . . . . 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 3331 . . . . . . 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 3331 . . . . . 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 3320 . . . . . . 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 3332 . . . . . 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 1434 . . . . 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 2738 . . . 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 1340 . 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 6903 . . . 4 (2nd β€˜π‘…) ∈ V
24 eleq1 2819 . . . 4 (𝐻 = (2nd β€˜π‘…) β†’ (𝐻 ∈ V ↔ (2nd β€˜π‘…) ∈ V))
2523, 24mpbiri 257 . . 3 (𝐻 = (2nd β€˜π‘…) β†’ 𝐻 ∈ V)
26 eqid 2730 . . . 4 dom dom 𝐻 = dom dom 𝐻
2726ismndo1 37044 . . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   Γ— cxp 5673  dom cdm 5675  ran crn 5676  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  AbelOpcablo 30064  MndOpcmndo 37037  RingOpscrngo 37065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7414  df-1st 7977  df-2nd 7978  df-grpo 30013  df-ablo 30065  df-ass 37014  df-exid 37016  df-mgmOLD 37020  df-sgrOLD 37032  df-mndo 37038  df-rngo 37066
This theorem is referenced by:  rngoidmlem  37107  rngo1cl  37110  isdrngo2  37129
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