Theorem genpassg 7610

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 7608	. 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 7609	. 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 6082	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
9	4	caovcl 6082	. . . . 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 1196	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
12	4	caovcl 6082	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
13	4	caovcl 6082	. . . . 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 1201	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A F ( B F C ) ) e. P. )
16		preqlu 7556	. . 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 946	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 980   = wceq 1364   e. wcel 2167   E.wrex 2476   {crab 2479   <.cop 3626   X. cxp 4662   dom cdm 4664   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   P.cnp 7375

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-qs 6607  df-ni 7388  df-nqqs 7432  df-inp 7550

This theorem is referenced by:  addassprg  7663  mulassprg  7665