Theorem addcomprg 7597

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 7494	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7500	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
3	1, 2	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
4		prop 7494	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7500	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
6	4, 5	sylan 283	. . . . . . . . . . . 12 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
7		addcomnqg 7400	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) = ( z +Q y ) )
8	7	eqeq2d 2201	. . . . . . . . . . . 12 |- ( ( y e. Q. /\ z e. Q. ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
9	6, 8	sylan2 286	. . . . . . . . . . 11 |- ( ( y e. Q. /\ ( A e. P. /\ z e. ( 1st ` A ) ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
10	9	anassrs 400	. . . . . . . . . 10 |- ( ( ( y e. Q. /\ A e. P. ) /\ z e. ( 1st ` A ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
11	10	rexbidva 2487	. . . . . . . . 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 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 ) ) )
13	3, 12	sylan2 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 ) ) )
14	13	anassrs 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 ) ) )
15	14	rexbidva 2487	. . . . 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 2654	. . . . 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 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 ) ) )
18	17	rabbidv 2741	. . 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 7501	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
20	1, 19	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
21		elprnqu 7501	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
22	4, 21	sylan 283	. . . . . . . . . . . 12 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
23	22, 8	sylan2 286	. . . . . . . . . . 11 |- ( ( y e. Q. /\ ( A e. P. /\ z e. ( 2nd ` A ) ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
24	23	anassrs 400	. . . . . . . . . 10 |- ( ( ( y e. Q. /\ A e. P. ) /\ z e. ( 2nd ` A ) ) -> ( x = ( y +Q z ) <-> x = ( z +Q y ) ) )
25	24	rexbidva 2487	. . . . . . . . 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 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 ) ) )
27	20, 26	sylan2 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 ) ) )
28	27	anassrs 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 ) ) )
29	28	rexbidva 2487	. . . . 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 2654	. . . . 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 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 ) ) )
32	31	rabbidv 2741	. . 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 3801	. 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 7557	. . 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 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 7557	. 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 2233	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = ( B +P. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   E.wrex 2469   {crab 2472   <.cop 3610   ` cfv 5232  (class class class)co 5892   1stc1st 6158   2ndc2nd 6159   Q.cnq 7299   +Q cplq 7301   P.cnp 7310   +P. cpp 7312

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7323  df-pli 7324  df-mi 7325  df-plpq 7363  df-enq 7366  df-nqqs 7367  df-plqqs 7368  df-inp 7485  df-iplp 7487

This theorem is referenced by:  prplnqu  7639  addextpr  7640  caucvgprlemcanl  7663  caucvgprprlemnkltj  7708  caucvgprprlemnbj  7712  caucvgprprlemmu  7714  caucvgprprlemloc  7722  caucvgprprlemexbt  7725  caucvgprprlemexb  7726  caucvgprprlemaddq  7727  enrer  7754  addcmpblnr  7758  mulcmpblnrlemg  7759  ltsrprg  7766  addcomsrg  7774  mulcomsrg  7776  mulasssrg  7777  distrsrg  7778  lttrsr  7781  ltposr  7782  ltsosr  7783  0lt1sr  7784  0idsr  7786  1idsr  7787  ltasrg  7789  recexgt0sr  7792  mulgt0sr  7797  aptisr  7798  mulextsr1lem  7799  archsr  7801  srpospr  7802  prsrpos  7804  prsradd  7805  prsrlt  7806  ltpsrprg  7822  map2psrprg  7824  pitonnlem1p1  7865  pitoregt0  7868  recidpirqlemcalc  7876