Theorem genpassg 7745

Description: Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)

Hypotheses

Ref	Expression
genpelvl.1	|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
genpelvl.2	|- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
genpassg.4	|- dom F = ( P. X. P. )
genpassg.5	|- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )
genpassg.6	|- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )

Assertion

Ref	Expression
genpassg	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Distinct variable groups:   x, y, z, f, g, h, w, v, A   x, B, y, z, f, g, h, w, v   x, G, y, z, f, g, h, w, v   f, F, g   C, f, g, h, v, w, x, y, z   h, F, v, w, x, y, z

Proof of Theorem genpassg

Step	Hyp	Ref	Expression
1		genpelvl.1	. . 3 |- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
2		genpelvl.2	. . 3 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
3		genpassg.4	. . 3 |- dom F = ( P. X. P. )
4		genpassg.5	. . 3 |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )
5		genpassg.6	. . 3 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )
6	1, 2, 3, 4, 5	genpassl 7743	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( ( A F B ) F C ) ) = ( 1st ` ( A F ( B F C ) ) ) )
7	1, 2, 3, 4, 5	genpassu 7744	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) )
8	4	caovcl 6176	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
9	4	caovcl 6176	. . . . 5 |- ( ( ( A F B ) e. P. /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
10	8, 9	sylan 283	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
11	10	3impa 1220	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
12	4	caovcl 6176	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
13	4	caovcl 6176	. . . . 5 |- ( ( A e. P. /\ ( B F C ) e. P. ) -> ( A F ( B F C ) ) e. P. )
14	12, 13	sylan2 286	. . . 4 |- ( ( A e. P. /\ ( B e. P. /\ C e. P. ) ) -> ( A F ( B F C ) ) e. P. )
15	14	3impb 1225	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A F ( B F C ) ) e. P. )
16		preqlu 7691	. . 3 |- ( ( ( ( A F B ) F C ) e. P. /\ ( A F ( B F C ) ) e. P. ) -> ( ( ( A F B ) F C ) = ( A F ( B F C ) ) <-> ( ( 1st ` ( ( A F B ) F C ) ) = ( 1st ` ( A F ( B F C ) ) ) /\ ( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) ) ) )
17	11, 15, 16	syl2anc 411	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( ( A F B ) F C ) = ( A F ( B F C ) ) <-> ( ( 1st ` ( ( A F B ) F C ) ) = ( 1st ` ( A F ( B F C ) ) ) /\ ( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) ) ) )
18	6, 7, 17	mpbir2and 952	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   {crab 2514   <.cop 3672   X. cxp 4723   dom cdm 4725   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   P.cnp 7510

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-qs 6707  df-ni 7523  df-nqqs 7567  df-inp 7685

This theorem is referenced by:  addassprg  7798  mulassprg  7800