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Theorem addcomprg 7350
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  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  ( B  +P.  A ) )

Proof of Theorem addcomprg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7247 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7253 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
31, 2sylan 279 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
4 prop 7247 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7253 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
64, 5sylan 279 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
7 addcomnqg 7153 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  =  ( z  +Q  y ) )
87eqeq2d 2127 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( x  =  ( y  +Q  z )  <-> 
x  =  ( z  +Q  y ) ) )
96, 8sylan2 282 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 1st `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
109anassrs 395 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 1st `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
1110rexbidva 2409 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  A  e.  P. )  ->  ( E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 1st `  A ) x  =  ( z  +Q  y ) ) )
1211ancoms 266 . . . . . . . 8  |-  ( ( A  e.  P.  /\  y  e.  Q. )  ->  ( E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 1st `  A ) x  =  ( z  +Q  y ) ) )
133, 12sylan2 282 . . . . . . 7  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  y  e.  ( 1st `  B ) ) )  ->  ( E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 1st `  A ) x  =  ( z  +Q  y ) ) )
1413anassrs 395 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  y  e.  ( 1st `  B ) )  ->  ( E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 1st `  A ) x  =  ( z  +Q  y ) ) )
1514rexbidva 2409 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. y  e.  ( 1st `  B
) E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z )  <->  E. y  e.  ( 1st `  B ) E. z  e.  ( 1st `  A ) x  =  ( z  +Q  y
) ) )
16 rexcom 2570 . . . . 5  |-  ( E. y  e.  ( 1st `  B ) E. z  e.  ( 1st `  A
) x  =  ( z  +Q  y )  <->  E. z  e.  ( 1st `  A ) E. y  e.  ( 1st `  B ) x  =  ( z  +Q  y
) )
1715, 16syl6bb 195 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. y  e.  ( 1st `  B
) E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 1st `  A ) E. y  e.  ( 1st `  B ) x  =  ( z  +Q  y
) ) )
1817rabbidv 2647 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  { x  e.  Q.  |  E. y  e.  ( 1st `  B ) E. z  e.  ( 1st `  A ) x  =  ( y  +Q  z ) }  =  { x  e. 
Q.  |  E. z  e.  ( 1st `  A
) E. y  e.  ( 1st `  B
) x  =  ( z  +Q  y ) } )
19 elprnqu 7254 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
201, 19sylan 279 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
21 elprnqu 7254 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
224, 21sylan 279 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
2322, 8sylan2 282 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 2nd `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2423anassrs 395 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 2nd `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2524rexbidva 2409 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  A  e.  P. )  ->  ( E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 2nd `  A ) x  =  ( z  +Q  y ) ) )
2625ancoms 266 . . . . . . . 8  |-  ( ( A  e.  P.  /\  y  e.  Q. )  ->  ( E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 2nd `  A ) x  =  ( z  +Q  y ) ) )
2720, 26sylan2 282 . . . . . . 7  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  y  e.  ( 2nd `  B ) ) )  ->  ( E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 2nd `  A ) x  =  ( z  +Q  y ) ) )
2827anassrs 395 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  y  e.  ( 2nd `  B ) )  ->  ( E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 2nd `  A ) x  =  ( z  +Q  y ) ) )
2928rexbidva 2409 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. y  e.  ( 2nd `  B
) E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z )  <->  E. y  e.  ( 2nd `  B ) E. z  e.  ( 2nd `  A ) x  =  ( z  +Q  y
) ) )
30 rexcom 2570 . . . . 5  |-  ( E. y  e.  ( 2nd `  B ) E. z  e.  ( 2nd `  A
) x  =  ( z  +Q  y )  <->  E. z  e.  ( 2nd `  A ) E. y  e.  ( 2nd `  B ) x  =  ( z  +Q  y
) )
3129, 30syl6bb 195 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. y  e.  ( 2nd `  B
) E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z )  <->  E. z  e.  ( 2nd `  A ) E. y  e.  ( 2nd `  B ) x  =  ( z  +Q  y
) ) )
3231rabbidv 2647 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  { x  e.  Q.  |  E. y  e.  ( 2nd `  B ) E. z  e.  ( 2nd `  A ) x  =  ( y  +Q  z ) }  =  { x  e. 
Q.  |  E. z  e.  ( 2nd `  A
) E. y  e.  ( 2nd `  B
) x  =  ( z  +Q  y ) } )
3318, 32opeq12d 3681 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. { x  e.  Q.  |  E. y  e.  ( 1st `  B ) E. z  e.  ( 1st `  A ) x  =  ( y  +Q  z ) } ,  { x  e. 
Q.  |  E. y  e.  ( 2nd `  B
) E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z ) } >.  =  <. { x  e.  Q.  |  E. z  e.  ( 1st `  A ) E. y  e.  ( 1st `  B ) x  =  ( z  +Q  y
) } ,  {
x  e.  Q.  |  E. z  e.  ( 2nd `  A ) E. y  e.  ( 2nd `  B ) x  =  ( z  +Q  y
) } >. )
34 plpvlu 7310 . . 3  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  +P.  A
)  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  B
) E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  B
) E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z ) } >. )
3534ancoms 266 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( B  +P.  A
)  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  B
) E. z  e.  ( 1st `  A
) x  =  ( y  +Q  z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  B
) E. z  e.  ( 2nd `  A
) x  =  ( y  +Q  z ) } >. )
36 plpvlu 7310 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  <. { x  e.  Q.  |  E. z  e.  ( 1st `  A
) E. y  e.  ( 1st `  B
) x  =  ( z  +Q  y ) } ,  { x  e.  Q.  |  E. z  e.  ( 2nd `  A
) E. y  e.  ( 2nd `  B
) x  =  ( z  +Q  y ) } >. )
3733, 35, 363eqtr4rd 2159 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  ( B  +P.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392   {crab 2395   <.cop 3498   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    +Q cplq 7054   P.cnp 7063    +P. cpp 7065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-plpq 7116  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-inp 7238  df-iplp 7240
This theorem is referenced by:  prplnqu  7392  addextpr  7393  caucvgprlemcanl  7416  caucvgprprlemnkltj  7461  caucvgprprlemnbj  7465  caucvgprprlemmu  7467  caucvgprprlemloc  7475  caucvgprprlemexbt  7478  caucvgprprlemexb  7479  caucvgprprlemaddq  7480  enrer  7507  addcmpblnr  7511  mulcmpblnrlemg  7512  ltsrprg  7519  addcomsrg  7527  mulcomsrg  7529  mulasssrg  7530  distrsrg  7531  lttrsr  7534  ltposr  7535  ltsosr  7536  0lt1sr  7537  0idsr  7539  1idsr  7540  ltasrg  7542  recexgt0sr  7545  mulgt0sr  7550  aptisr  7551  mulextsr1lem  7552  archsr  7554  srpospr  7555  prsrpos  7557  prsradd  7558  prsrlt  7559  ltpsrprg  7575  map2psrprg  7577  pitonnlem1p1  7618  pitoregt0  7621  recidpirqlemcalc  7629
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