Theorem addcomprg 7519

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.

Step	Hyp	Ref	Expression
1		prop 7416	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7422	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
3	1, 2	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
4		prop 7416	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7422	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
6	4, 5	sylan 281	. . . . . . . . . . . 12 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
7		addcomnqg 7322	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) = ( z +Q y ) )
8	7	eqeq2d 2177	. . . . . . . . . . . 12 |- ( ( y e. Q. /\ z e. Q. ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
9	6, 8	sylan2 284	. . . . . . . . . . 11 |- ( ( y e. Q. /\ ( A e. P. /\ z e. ( 1st ` A ) ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
10	9	anassrs 398	. . . . . . . . . 10 |- ( ( ( y e. Q. /\ A e. P. ) /\ z e. ( 1st ` A ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
11	10	rexbidva 2463	. . . . . . . . 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 ) ) )
12	11	ancoms 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 ) ) )
13	3, 12	sylan2 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 ) ) )
14	13	anassrs 398	. . . . . 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 ) ) )
15	14	rexbidva 2463	. . . . 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 2630	. . . . 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 ) )
17	15, 16	bitrdi 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 ) ) )
18	17	rabbidv 2715	. . 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 7423	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
20	1, 19	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
21		elprnqu 7423	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
22	4, 21	sylan 281	. . . . . . . . . . . 12 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
23	22, 8	sylan2 284	. . . . . . . . . . 11 |- ( ( y e. Q. /\ ( A e. P. /\ z e. ( 2nd ` A ) ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
24	23	anassrs 398	. . . . . . . . . 10 |- ( ( ( y e. Q. /\ A e. P. ) /\ z e. ( 2nd ` A ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
25	24	rexbidva 2463	. . . . . . . . 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 ) ) )
26	25	ancoms 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 ) ) )
27	20, 26	sylan2 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 ) ) )
28	27	anassrs 398	. . . . . 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 ) ) )
29	28	rexbidva 2463	. . . . 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 2630	. . . . 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 ) )
31	29, 30	bitrdi 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 ) ) )
32	31	rabbidv 2715	. . 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 ) } )
33	18, 32	opeq12d 3766	. 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 7479	. . 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 ) } >. )
35	34	ancoms 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 7479	. 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 ) } >. )
37	33, 35, 36	3eqtr4rd 2209	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = ( B +P. A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   {crab 2448   <.cop 3579   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   P.cnp 7232   +P. cpp 7234

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-plpq 7285  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-inp 7407  df-iplp 7409

This theorem is referenced by:  prplnqu  7561  addextpr  7562  caucvgprlemcanl  7585  caucvgprprlemnkltj  7630  caucvgprprlemnbj  7634  caucvgprprlemmu  7636  caucvgprprlemloc  7644  caucvgprprlemexbt  7647  caucvgprprlemexb  7648  caucvgprprlemaddq  7649  enrer  7676  addcmpblnr  7680  mulcmpblnrlemg  7681  ltsrprg  7688  addcomsrg  7696  mulcomsrg  7698  mulasssrg  7699  distrsrg  7700  lttrsr  7703  ltposr  7704  ltsosr  7705  0lt1sr  7706  0idsr  7708  1idsr  7709  ltasrg  7711  recexgt0sr  7714  mulgt0sr  7719  aptisr  7720  mulextsr1lem  7721  archsr  7723  srpospr  7724  prsrpos  7726  prsradd  7727  prsrlt  7728  ltpsrprg  7744  map2psrprg  7746  pitonnlem1p1  7787  pitoregt0  7790  recidpirqlemcalc  7798