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Theorem addcomprg 7577
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 7474 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7480 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
31, 2sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
4 prop 7474 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7480 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
64, 5sylan 283 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
7 addcomnqg 7380 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  =  ( z  +Q  y ) )
87eqeq2d 2189 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( x  =  ( y  +Q  z )  <-> 
x  =  ( z  +Q  y ) ) )
96, 8sylan2 286 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 1st `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
109anassrs 400 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 1st `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
1110rexbidva 2474 . . . . . . . . 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 268 . . . . . . . 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 286 . . . . . . 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 400 . . . . . 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 2474 . . . . 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 2641 . . . . 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, 16bitrdi 196 . . . 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 2727 . . 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 7481 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
201, 19sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
21 elprnqu 7481 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
224, 21sylan 283 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
2322, 8sylan2 286 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 2nd `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2423anassrs 400 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 2nd `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2524rexbidva 2474 . . . . . . . . 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 268 . . . . . . . 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 286 . . . . . . 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 400 . . . . . 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 2474 . . . . 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 2641 . . . . 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, 30bitrdi 196 . . . 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 2727 . . 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 3787 . 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 7537 . . 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 268 . 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 7537 . 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 2221 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  ( B  +P.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3596   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279    +Q cplq 7281   P.cnp 7290    +P. cpp 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-plpq 7343  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-inp 7465  df-iplp 7467
This theorem is referenced by:  prplnqu  7619  addextpr  7620  caucvgprlemcanl  7643  caucvgprprlemnkltj  7688  caucvgprprlemnbj  7692  caucvgprprlemmu  7694  caucvgprprlemloc  7702  caucvgprprlemexbt  7705  caucvgprprlemexb  7706  caucvgprprlemaddq  7707  enrer  7734  addcmpblnr  7738  mulcmpblnrlemg  7739  ltsrprg  7746  addcomsrg  7754  mulcomsrg  7756  mulasssrg  7757  distrsrg  7758  lttrsr  7761  ltposr  7762  ltsosr  7763  0lt1sr  7764  0idsr  7766  1idsr  7767  ltasrg  7769  recexgt0sr  7772  mulgt0sr  7777  aptisr  7778  mulextsr1lem  7779  archsr  7781  srpospr  7782  prsrpos  7784  prsradd  7785  prsrlt  7786  ltpsrprg  7802  map2psrprg  7804  pitonnlem1p1  7845  pitoregt0  7848  recidpirqlemcalc  7856
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