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Theorem addcomprg 7354
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 7251 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7257 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
31, 2sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
4 prop 7251 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7257 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
64, 5sylan 281 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
7 addcomnqg 7157 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  =  ( z  +Q  y ) )
87eqeq2d 2129 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( x  =  ( y  +Q  z )  <-> 
x  =  ( z  +Q  y ) ) )
96, 8sylan2 284 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 1st `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
109anassrs 397 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 1st `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
1110rexbidva 2411 . . . . . . . . 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 284 . . . . . . 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 397 . . . . . 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 2411 . . . . 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 2572 . . . . 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 2649 . . 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 7258 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
201, 19sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
21 elprnqu 7258 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
224, 21sylan 281 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
2322, 8sylan2 284 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  ( A  e.  P.  /\  z  e.  ( 2nd `  A ) ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2423anassrs 397 . . . . . . . . . 10  |-  ( ( ( y  e.  Q.  /\  A  e.  P. )  /\  z  e.  ( 2nd `  A ) )  ->  ( x  =  ( y  +Q  z
)  <->  x  =  (
z  +Q  y ) ) )
2524rexbidva 2411 . . . . . . . . 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 284 . . . . . . 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 397 . . . . . 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 2411 . . . . 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 2572 . . . . 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 2649 . . 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 3683 . 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 7314 . . 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 7314 . 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 2161 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 1316    e. wcel 1465   E.wrex 2394   {crab 2397   <.cop 3500   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058   P.cnp 7067    +P. cpp 7069
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-plpq 7120  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-inp 7242  df-iplp 7244
This theorem is referenced by:  prplnqu  7396  addextpr  7397  caucvgprlemcanl  7420  caucvgprprlemnkltj  7465  caucvgprprlemnbj  7469  caucvgprprlemmu  7471  caucvgprprlemloc  7479  caucvgprprlemexbt  7482  caucvgprprlemexb  7483  caucvgprprlemaddq  7484  enrer  7511  addcmpblnr  7515  mulcmpblnrlemg  7516  ltsrprg  7523  addcomsrg  7531  mulcomsrg  7533  mulasssrg  7534  distrsrg  7535  lttrsr  7538  ltposr  7539  ltsosr  7540  0lt1sr  7541  0idsr  7543  1idsr  7544  ltasrg  7546  recexgt0sr  7549  mulgt0sr  7554  aptisr  7555  mulextsr1lem  7556  archsr  7558  srpospr  7559  prsrpos  7561  prsradd  7562  prsrlt  7563  ltpsrprg  7579  map2psrprg  7581  pitonnlem1p1  7622  pitoregt0  7625  recidpirqlemcalc  7633
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