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Theorem rngomndo 36444
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 2733 . . . 4 (1st β€˜π‘…) = (1st β€˜π‘…)
2 unmnd.1 . . . 4 𝐻 = (2nd β€˜π‘…)
3 eqid 2733 . . . 4 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3rngosm 36409 . . 3 (𝑅 ∈ RingOps β†’ 𝐻:(ran (1st β€˜π‘…) Γ— ran (1st β€˜π‘…))⟢ran (1st β€˜π‘…))
51, 2, 3rngoass 36415 . . . 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 3197 . . 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 36408 . . . 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 773 . . 3 (𝑅 ∈ RingOps β†’ βˆƒπ‘₯ ∈ ran (1st β€˜π‘…)βˆ€π‘¦ ∈ ran (1st β€˜π‘…)((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
92, 1rngorn1 36442 . . . 4 (𝑅 ∈ RingOps β†’ ran (1st β€˜π‘…) = dom dom 𝐻)
10 xpid11 5891 . . . . . . . 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 6656 . . . . . . 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 687 . . . . . 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 3308 . . . . . . . 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 3306 . . . . . . 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 3306 . . . . . 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 3308 . . . . . . 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 3307 . . . . . 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 1437 . . . . 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 2741 . . . 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 1343 . 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 6859 . . . 4 (2nd β€˜π‘…) ∈ V
24 eleq1 2822 . . . 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 2733 . . . 4 dom dom 𝐻 = dom dom 𝐻
2726ismndo1 36382 . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   Γ— cxp 5635  dom cdm 5637  ran crn 5638  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  AbelOpcablo 29535  MndOpcmndo 36375  RingOpscrngo 36403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-ov 7364  df-1st 7925  df-2nd 7926  df-grpo 29484  df-ablo 29536  df-ass 36352  df-exid 36354  df-mgmOLD 36358  df-sgrOLD 36370  df-mndo 36376  df-rngo 36404
This theorem is referenced by:  rngoidmlem  36445  rngo1cl  36448  isdrngo2  36467
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