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Theorem addcomprg 6553
Description: Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
Assertion
Ref Expression
addcomprg  P.  P.  +P.  +P.

Proof of Theorem addcomprg
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 addcomnqg 6365 . . . . . . . . . . . . 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 plpvlu 6520 . . 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 plpvlu 6520 . 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 cplq 6266   P.cnp 6275    +P. cpp 6277
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-pli 6289  df-mi 6290  df-plpq 6328  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-inp 6448  df-iplp 6450
This theorem is referenced by:  addextpr  6592  cauappcvgprlemcan  6615  enrer  6643  addcmpblnr  6647  mulcmpblnrlemg  6648  ltsrprg  6655  addcomsrg  6663  mulcomsrg  6665  mulasssrg  6666  distrsrg  6667  lttrsr  6670  ltposr  6671  ltsosr  6672  0lt1sr  6673  0idsr  6675  1idsr  6676  ltasrg  6678  recexgt0sr  6681  mulgt0sr  6684  aptisr  6685  mulextsr1lem  6686  archsr  6688  pitonnlem1p1  6722
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