Theorem addcomprg 7638

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 7535	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7541	. . . . . . . . 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 7535	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7541	. . . . . . . . . . . . 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 7441	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) = ( z +Q y ) )
8	7	eqeq2d 2205	. . . . . . . . . . . 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 2491	. . . . . . . . 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 2491	. . . . 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 2658	. . . . 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 2749	. . 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 7542	. . . . . . . . 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 7542	. . . . . . . . . . . . 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 2491	. . . . . . . . 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 2491	. . . . 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 2658	. . . . 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 2749	. . 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 3812	. 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 7598	. . 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 7598	. 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 2237	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 2164   E.wrex 2473   {crab 2476   <.cop 3621   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   +Q cplq 7342   P.cnp 7351   +P. cpp 7353

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-plpq 7404  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-inp 7526  df-iplp 7528

This theorem is referenced by:  prplnqu  7680  addextpr  7681  caucvgprlemcanl  7704  caucvgprprlemnkltj  7749  caucvgprprlemnbj  7753  caucvgprprlemmu  7755  caucvgprprlemloc  7763  caucvgprprlemexbt  7766  caucvgprprlemexb  7767  caucvgprprlemaddq  7768  enrer  7795  addcmpblnr  7799  mulcmpblnrlemg  7800  ltsrprg  7807  addcomsrg  7815  mulcomsrg  7817  mulasssrg  7818  distrsrg  7819  lttrsr  7822  ltposr  7823  ltsosr  7824  0lt1sr  7825  0idsr  7827  1idsr  7828  ltasrg  7830  recexgt0sr  7833  mulgt0sr  7838  aptisr  7839  mulextsr1lem  7840  archsr  7842  srpospr  7843  prsrpos  7845  prsradd  7846  prsrlt  7847  ltpsrprg  7863  map2psrprg  7865  pitonnlem1p1  7906  pitoregt0  7909  recidpirqlemcalc  7917