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Theorem rngomndo 25099
 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
Assertion
Ref Expression
rngomndo MndOp

Proof of Theorem rngomndo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4
2 unmnd.1 . . . 4
3 eqid 2467 . . . 4
41, 2, 3rngosm 25059 . . 3
51, 2, 3rngoass 25065 . . . 4
65ralrimivvva 2886 . . 3
71, 2, 3rngoi 25058 . . . . 5
87simprd 463 . . . 4
98simprd 463 . . 3
102, 1rngorn1 25097 . . . 4
11 xpid11 5222 . . . . . . . 8
1211biimpri 206 . . . . . . 7
13 feq23 5714 . . . . . . 7
1412, 13mpancom 669 . . . . . 6
15 raleq 3058 . . . . . . . 8
1615raleqbi1dv 3066 . . . . . . 7
1716raleqbi1dv 3066 . . . . . 6
18 raleq 3058 . . . . . . 7
1918rexeqbi1dv 3067 . . . . . 6
2014, 17, 193anbi123d 1299 . . . . 5
2120eqcoms 2479 . . . 4
2210, 21syl 16 . . 3
234, 6, 9, 22mpbir3and 1179 . 2
24 fvex 5874 . . . 4
25 eleq1 2539 . . . 4
2624, 25mpbiri 233 . . 3
27 eqid 2467 . . . 4
2827ismndo1 25022 . . 3 MndOp
292, 26, 28mp2b 10 . 2 MndOp
3023, 29sylibr 212 1 MndOp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2814  wrex 2815  cvv 3113   cxp 4997   cdm 4999   crn 5000  wf 5582  cfv 5586  (class class class)co 6282  c1st 6779  c2nd 6780  cablo 24959  MndOpcmndo 25015  crngo 25053 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-ov 6285  df-1st 6781  df-2nd 6782  df-grpo 24869  df-ablo 24960  df-ass 24991  df-exid 24993  df-mgm 24997  df-sgr 25009  df-mndo 25016  df-rngo 25054 This theorem is referenced by:  rngoidmlem  25101  rngo1cl  25107  isdrngo2  29964
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