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Theorem rngomndo 35835
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 2737 . . . 4 (1st𝑅) = (1st𝑅)
2 unmnd.1 . . . 4 𝐻 = (2nd𝑅)
3 eqid 2737 . . . 4 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3rngosm 35800 . . 3 (𝑅 ∈ RingOps → 𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅))
51, 2, 3rngoass 35806 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
65ralrimivvva 3113 . . 3 (𝑅 ∈ RingOps → ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
71, 2, 3rngoi 35799 . . . 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 774 . . 3 (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
92, 1rngorn1 35833 . . . 4 (𝑅 ∈ RingOps → ran (1st𝑅) = dom dom 𝐻)
10 xpid11 5806 . . . . . . . 8 ((dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)) ↔ dom dom 𝐻 = ran (1st𝑅))
1110biimpri 231 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)))
12 feq23 6534 . . . . . . 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 688 . . . . . 6 (dom dom 𝐻 = ran (1st𝑅) → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅)))
14 raleq 3324 . . . . . . . 8 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
1514raleqbi1dv 3322 . . . . . . 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 3322 . . . . . 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 3324 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
1817rexeqbi1dv 3323 . . . . . 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 1438 . . . . 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 2745 . . . 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 1344 . 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 6735 . . . 4 (2nd𝑅) ∈ V
24 eleq1 2825 . . . 4 (𝐻 = (2nd𝑅) → (𝐻 ∈ V ↔ (2nd𝑅) ∈ V))
2523, 24mpbiri 261 . . 3 (𝐻 = (2nd𝑅) → 𝐻 ∈ V)
26 eqid 2737 . . . 4 dom dom 𝐻 = dom dom 𝐻
2726ismndo1 35773 . . 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 237 1 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  Vcvv 3413   × cxp 5554  dom cdm 5556  ran crn 5557  wf 6381  cfv 6385  (class class class)co 7218  1st c1st 7764  2nd c2nd 7765  AbelOpcablo 28630  MndOpcmndo 35766  RingOpscrngo 35794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-fo 6391  df-fv 6393  df-ov 7221  df-1st 7766  df-2nd 7767  df-grpo 28579  df-ablo 28631  df-ass 35743  df-exid 35745  df-mgmOLD 35749  df-sgrOLD 35761  df-mndo 35767  df-rngo 35795
This theorem is referenced by:  rngoidmlem  35836  rngo1cl  35839  isdrngo2  35858
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