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Theorem mulcomprg 6555
Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
Assertion
Ref Expression
mulcomprg  P.  P.  .P.  .P.

Proof of Theorem mulcomprg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6457 . . . . . . . . 9  P.  <. 1st `  ,  2nd `  >.  P.
2 elprnql 6463 . . . . . . . . 9 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
31, 2sylan 267 . . . . . . . 8  P.  1st `  Q.
4 prop 6457 . . . . . . . . . . . . 13  P.  <. 1st `  ,  2nd `  >.  P.
5 elprnql 6463 . . . . . . . . . . . . 13 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
64, 5sylan 267 . . . . . . . . . . . 12  P.  1st `  Q.
7 mulcomnqg 6367 . . . . . . . . . . . . 13  Q.  Q.  .Q  .Q
87eqeq2d 2048 . . . . . . . . . . . 12  Q.  Q.  .Q  .Q
96, 8sylan2 270 . . . . . . . . . . 11  Q.  P.  1st `  .Q  .Q
109anassrs 380 . . . . . . . . . 10  Q.  P.  1st `  .Q  .Q
1110rexbidva 2317 . . . . . . . . 9  Q.  P.  1st `  .Q  1st `  .Q
1211ancoms 255 . . . . . . . 8  P.  Q.  1st `  .Q  1st `  .Q
133, 12sylan2 270 . . . . . . 7  P.  P.  1st `  1st `  .Q  1st `  .Q
1413anassrs 380 . . . . . 6  P.  P.  1st `  1st `  .Q  1st `  .Q
1514rexbidva 2317 . . . . 5  P.  P.  1st `  1st `  .Q  1st `  1st `  .Q
16 rexcom 2468 . . . . 5  1st `  1st `  .Q  1st `  1st `  .Q
1715, 16syl6bb 185 . . . 4  P.  P.  1st `  1st `  .Q  1st `  1st `  .Q
1817rabbidv 2543 . . 3  P.  P.  {  Q.  |  1st `  1st `  .Q  }  {  Q.  |  1st `  1st `  .Q  }
19 elprnqu 6464 . . . . . . . . 9 
<. 1st `  ,  2nd `  >.  P.  2nd `  Q.
201, 19sylan 267 . . . . . . . 8  P.  2nd `  Q.
21 elprnqu 6464 . . . . . . . . . . . . 13 
<. 1st `  ,  2nd `  >.  P.  2nd `  Q.
224, 21sylan 267 . . . . . . . . . . . 12  P.  2nd `  Q.
2322, 8sylan2 270 . . . . . . . . . . 11  Q.  P.  2nd `  .Q  .Q
2423anassrs 380 . . . . . . . . . 10  Q.  P.  2nd `  .Q  .Q
2524rexbidva 2317 . . . . . . . . 9  Q.  P.  2nd `  .Q  2nd `  .Q
2625ancoms 255 . . . . . . . 8  P.  Q.  2nd `  .Q  2nd `  .Q
2720, 26sylan2 270 . . . . . . 7  P.  P.  2nd `  2nd `  .Q  2nd `  .Q
2827anassrs 380 . . . . . 6  P.  P.  2nd `  2nd `  .Q  2nd `  .Q
2928rexbidva 2317 . . . . 5  P.  P.  2nd `  2nd `  .Q  2nd `  2nd `  .Q
30 rexcom 2468 . . . . 5  2nd `  2nd `  .Q  2nd `  2nd `  .Q
3129, 30syl6bb 185 . . . 4  P.  P.  2nd `  2nd `  .Q  2nd `  2nd `  .Q
3231rabbidv 2543 . . 3  P.  P.  {  Q.  |  2nd `  2nd `  .Q  }  {  Q.  |  2nd `  2nd `  .Q  }
3318, 32opeq12d 3548 . 2  P.  P.  <. { 
Q.  |  1st `  1st `  .Q  } ,  {  Q.  |  2nd `  2nd `  .Q  } >.  <. {  Q.  |  1st `  1st `  .Q  } ,  {  Q.  |  2nd `  2nd `  .Q  } >.
34 mpvlu 6521 . . 3  P.  P.  .P.  <. {  Q.  |  1st `  1st `  .Q  } ,  {  Q.  |  2nd `  2nd `  .Q  } >.
3534ancoms 255 . 2  P.  P.  .P.  <. {  Q.  |  1st `  1st `  .Q  } ,  {  Q.  |  2nd `  2nd `  .Q  } >.
36 mpvlu 6521 . 2  P.  P.  .P.  <. {  Q.  |  1st `  1st `  .Q  } ,  {  Q.  |  2nd `  2nd `  .Q  } >.
3733, 35, 363eqtr4rd 2080 1  P.  P.  .P.  .P.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wrex 2301   {crab 2304   <.cop 3370   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264    .Q cmq 6267   P.cnp 6275    .P. cmp 6278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-mqqs 6334  df-inp 6448  df-imp 6451
This theorem is referenced by:  ltmprr  6613  mulcmpblnrlemg  6648  mulcomsrg  6665  mulasssrg  6666  m1m1sr  6669  recexgt0sr  6681  mulgt0sr  6684  mulextsr1lem  6686
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