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Theorem rngomndo 33393
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 2621 . . . 4 (1st𝑅) = (1st𝑅)
2 unmnd.1 . . . 4 𝐻 = (2nd𝑅)
3 eqid 2621 . . . 4 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3rngosm 33358 . . 3 (𝑅 ∈ RingOps → 𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅))
51, 2, 3rngoass 33364 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
65ralrimivvva 2967 . . 3 (𝑅 ∈ RingOps → ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
71, 2, 3rngoi 33357 . . . 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 796 . . 3 (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
92, 1rngorn1 33391 . . . 4 (𝑅 ∈ RingOps → ran (1st𝑅) = dom dom 𝐻)
10 xpid11 5312 . . . . . . . 8 ((dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)) ↔ dom dom 𝐻 = ran (1st𝑅))
1110biimpri 218 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1st𝑅) × ran (1st𝑅)))
12 feq23 5991 . . . . . . 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 702 . . . . . 6 (dom dom 𝐻 = ran (1st𝑅) → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻𝐻:(ran (1st𝑅) × ran (1st𝑅))⟶ran (1st𝑅)))
14 raleq 3130 . . . . . . . 8 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑧 ∈ ran (1st𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
1514raleqbi1dv 3138 . . . . . . 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 3138 . . . . . 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 3130 . . . . . . 7 (dom dom 𝐻 = ran (1st𝑅) → (∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ ran (1st𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
1817rexeqbi1dv 3139 . . . . . 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 1396 . . . . 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 2629 . . . 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 1243 . 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 6163 . . . 4 (2nd𝑅) ∈ V
24 eleq1 2686 . . . 4 (𝐻 = (2nd𝑅) → (𝐻 ∈ V ↔ (2nd𝑅) ∈ V))
2523, 24mpbiri 248 . . 3 (𝐻 = (2nd𝑅) → 𝐻 ∈ V)
26 eqid 2621 . . . 4 dom dom 𝐻 = dom dom 𝐻
2726ismndo1 33331 . . 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 224 1 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189   × cxp 5077  dom cdm 5079  ran crn 5080  wf 5848  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  AbelOpcablo 27265  MndOpcmndo 33324  RingOpscrngo 33352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-ov 6613  df-1st 7120  df-2nd 7121  df-grpo 27214  df-ablo 27266  df-ass 33301  df-exid 33303  df-mgmOLD 33307  df-sgrOLD 33319  df-mndo 33325  df-rngo 33353
This theorem is referenced by:  rngoidmlem  33394  rngo1cl  33397  isdrngo2  33416
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