Theorem genpassg 7713

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 7711	. 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 7712	. 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 6160	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
9	4	caovcl 6160	. . . . 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 1218	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
12	4	caovcl 6160	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
13	4	caovcl 6160	. . . . 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 1223	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A F ( B F C ) ) e. P. )
16		preqlu 7659	. . 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 950	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 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   {crab 2512   <.cop 3669   X. cxp 4717   dom cdm 4719   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   P.cnp 7478

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-qs 6686  df-ni 7491  df-nqqs 7535  df-inp 7653

This theorem is referenced by:  addassprg  7766  mulassprg  7768