Theorem addcomprg 7803

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 7700	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7706	. . . . . . . . 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 7700	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7706	. . . . . . . . . . . . 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 7606	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) = ( z +Q y ) )
8	7	eqeq2d 2242	. . . . . . . . . . . 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 2528	. . . . . . . . 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 2528	. . . . 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 2696	. . . . 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 2790	. . 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 7707	. . . . . . . . 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 7707	. . . . . . . . . . . . 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 2528	. . . . . . . . 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 2528	. . . . 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 2696	. . . . 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 2790	. . 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 3871	. 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 7763	. . 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 7763	. 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 2274	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = ( B +P. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2201   E.wrex 2510   {crab 2513   <.cop 3673   ` cfv 5328  (class class class)co 6023   1stc1st 6306   2ndc2nd 6307   Q.cnq 7505   +Q cplq 7507   P.cnp 7516   +P. cpp 7518

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-oadd 6591  df-omul 6592  df-er 6707  df-ec 6709  df-qs 6713  df-ni 7529  df-pli 7530  df-mi 7531  df-plpq 7569  df-enq 7572  df-nqqs 7573  df-plqqs 7574  df-inp 7691  df-iplp 7693

This theorem is referenced by:  prplnqu  7845  addextpr  7846  caucvgprlemcanl  7869  caucvgprprlemnkltj  7914  caucvgprprlemnbj  7918  caucvgprprlemmu  7920  caucvgprprlemloc  7928  caucvgprprlemexbt  7931  caucvgprprlemexb  7932  caucvgprprlemaddq  7933  enrer  7960  addcmpblnr  7964  mulcmpblnrlemg  7965  ltsrprg  7972  addcomsrg  7980  mulcomsrg  7982  mulasssrg  7983  distrsrg  7984  lttrsr  7987  ltposr  7988  ltsosr  7989  0lt1sr  7990  0idsr  7992  1idsr  7993  ltasrg  7995  recexgt0sr  7998  mulgt0sr  8003  aptisr  8004  mulextsr1lem  8005  archsr  8007  srpospr  8008  prsrpos  8010  prsradd  8011  prsrlt  8012  ltpsrprg  8028  map2psrprg  8030  pitonnlem1p1  8071  pitoregt0  8074  recidpirqlemcalc  8082