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Theorem addcomprg 7576
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 7473 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7479 . . . . . . . . 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 7473 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7479 . . . . . . . . . . . . 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 7379 . . . . . . . . . . . . 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 2726 . . 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 7480 . . . . . . . . 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 7480 . . . . . . . . . . . . 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 2726 . . 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 3786 . 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 7536 . . 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 7536 . 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 3595   ` cfv 5216  (class class class)co 5874   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278    +Q cplq 7280   P.cnp 7289    +P. cpp 7291
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-plpq 7342  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-inp 7464  df-iplp 7466
This theorem is referenced by:  prplnqu  7618  addextpr  7619  caucvgprlemcanl  7642  caucvgprprlemnkltj  7687  caucvgprprlemnbj  7691  caucvgprprlemmu  7693  caucvgprprlemloc  7701  caucvgprprlemexbt  7704  caucvgprprlemexb  7705  caucvgprprlemaddq  7706  enrer  7733  addcmpblnr  7737  mulcmpblnrlemg  7738  ltsrprg  7745  addcomsrg  7753  mulcomsrg  7755  mulasssrg  7756  distrsrg  7757  lttrsr  7760  ltposr  7761  ltsosr  7762  0lt1sr  7763  0idsr  7765  1idsr  7766  ltasrg  7768  recexgt0sr  7771  mulgt0sr  7776  aptisr  7777  mulextsr1lem  7778  archsr  7780  srpospr  7781  prsrpos  7783  prsradd  7784  prsrlt  7785  ltpsrprg  7801  map2psrprg  7803  pitonnlem1p1  7844  pitoregt0  7847  recidpirqlemcalc  7855
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