Theorem addcomprg 7591

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 7488	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7494	. . . . . . . . 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 7488	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7494	. . . . . . . . . . . . 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 7394	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) = ( z +Q y ) )
8	7	eqeq2d 2199	. . . . . . . . . . . 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 2484	. . . . . . . . 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 2484	. . . . 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 2651	. . . . 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 2738	. . 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 7495	. . . . . . . . 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 7495	. . . . . . . . . . . . 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 2484	. . . . . . . . 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 2484	. . . . 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 2651	. . . . 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 2738	. . 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 3798	. 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 7551	. . 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 7551	. 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 2231	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = ( B +P. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   E.wrex 2466   {crab 2469   <.cop 3607   ` cfv 5228  (class class class)co 5888   1stc1st 6153   2ndc2nd 6154   Q.cnq 7293   +Q cplq 7295   P.cnp 7304   +P. cpp 7306

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-plpq 7357  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-inp 7479  df-iplp 7481

This theorem is referenced by:  prplnqu  7633  addextpr  7634  caucvgprlemcanl  7657  caucvgprprlemnkltj  7702  caucvgprprlemnbj  7706  caucvgprprlemmu  7708  caucvgprprlemloc  7716  caucvgprprlemexbt  7719  caucvgprprlemexb  7720  caucvgprprlemaddq  7721  enrer  7748  addcmpblnr  7752  mulcmpblnrlemg  7753  ltsrprg  7760  addcomsrg  7768  mulcomsrg  7770  mulasssrg  7771  distrsrg  7772  lttrsr  7775  ltposr  7776  ltsosr  7777  0lt1sr  7778  0idsr  7780  1idsr  7781  ltasrg  7783  recexgt0sr  7786  mulgt0sr  7791  aptisr  7792  mulextsr1lem  7793  archsr  7795  srpospr  7796  prsrpos  7798  prsradd  7799  prsrlt  7800  ltpsrprg  7816  map2psrprg  7818  pitonnlem1p1  7859  pitoregt0  7862  recidpirqlemcalc  7870