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Theorem rngomndo 21104
 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 2296 . . . 4
2 unmnd.1 . . . 4
3 eqid 2296 . . . 4
41, 2, 3rngosm 21064 . . 3
51, 2, 3rngoass 21070 . . . 4
65ralrimivvva 2649 . . 3
71, 2, 3rngoi 21063 . . . . 5
87simprd 449 . . . 4
98simprd 449 . . 3
102, 1rngorn1 21102 . . . 4
11 xpid11 4916 . . . . . . . 8
1211biimpri 197 . . . . . . 7
13 feq23 5394 . . . . . . 7
1412, 13mpancom 650 . . . . . 6
15 raleq 2749 . . . . . . . 8
1615raleqbi1dv 2757 . . . . . . 7
1716raleqbi1dv 2757 . . . . . 6
18 raleq 2749 . . . . . . 7
1918rexeqbi1dv 2758 . . . . . 6
2014, 17, 193anbi123d 1252 . . . . 5
2120eqcoms 2299 . . . 4
2210, 21syl 15 . . 3
234, 6, 9, 22mpbir3and 1135 . 2
24 fvex 5555 . . . 4
25 eleq1 2356 . . . 4
2624, 25mpbiri 224 . . 3
27 eqid 2296 . . . 4
2827ismndo1 21027 . . 3 MndOp
292, 26, 28mp2b 9 . 2 MndOp
3023, 29sylibr 203 1 MndOp
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934   wceq 1632   wcel 1696  wral 2556  wrex 2557  cvv 2801   cxp 4703   cdm 4705   crn 4706  wf 5267  cfv 5271  (class class class)co 5874  c1st 6136  c2nd 6137  cablo 20964  MndOpcmndo 21020  crngo 21058 This theorem is referenced by:  rngoidmlem  21106  rngo1cl  21112  ununr  25523  glmrngo  25585  isdrngo2  26692 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-grpo 20874  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059
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