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Theorem addcomprg 6734
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 6631 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 6637 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
31, 2sylan 271 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
4 prop 6631 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 6637 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
64, 5sylan 271 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
7 addcomnqg 6537 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  =  ( z  +Q  y ) )
87eqeq2d 2067 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( x  =  ( y  +Q  z )  <-> 
x  =  ( z  +Q  y ) ) )
96, 8sylan2 274 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 1st `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
109anassrs 386 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 1st `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
1110rexbidva 2340 . . . . . . . . 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 259 . . . . . . . 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 274 . . . . . . 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 386 . . . . . 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 2340 . . . . 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 2491 . . . . 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 189 . . . 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 2566 . . 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 6638 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
201, 19sylan 271 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
21 elprnqu 6638 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
224, 21sylan 271 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
2322, 8sylan2 274 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 2nd `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2423anassrs 386 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 2nd `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2524rexbidva 2340 . . . . . . . . 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 259 . . . . . . . 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 274 . . . . . . 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 386 . . . . . 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 2340 . . . . 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 2491 . . . . 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 189 . . . 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 2566 . . 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 3585 . 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 6694 . . 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 259 . 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 6694 . 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 2099 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  ( B  +P.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   E.wrex 2324   {crab 2327   <.cop 3406   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438   P.cnp 6447    +P. cpp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-plpq 6500  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-inp 6622  df-iplp 6624
This theorem is referenced by:  prplnqu  6776  addextpr  6777  caucvgprlemcanl  6800  caucvgprprlemnkltj  6845  caucvgprprlemnbj  6849  caucvgprprlemmu  6851  caucvgprprlemloc  6859  caucvgprprlemexbt  6862  caucvgprprlemexb  6863  caucvgprprlemaddq  6864  enrer  6878  addcmpblnr  6882  mulcmpblnrlemg  6883  ltsrprg  6890  addcomsrg  6898  mulcomsrg  6900  mulasssrg  6901  distrsrg  6902  lttrsr  6905  ltposr  6906  ltsosr  6907  0lt1sr  6908  0idsr  6910  1idsr  6911  ltasrg  6913  recexgt0sr  6916  mulgt0sr  6920  aptisr  6921  mulextsr1lem  6922  archsr  6924  srpospr  6925  prsrpos  6927  prsradd  6928  prsrlt  6929  pitonnlem1p1  6980  pitoregt0  6983  recidpirqlemcalc  6991
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