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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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