Theorem genpassg 7488

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 7486	. 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 7487	. 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 6007	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
9	4	caovcl 6007	. . . . 5 |- ( ( ( A F B ) e. P. /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
10	8, 9	sylan 281	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
11	10	3impa 1189	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A F B ) F C ) e. P. )
12	4	caovcl 6007	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
13	4	caovcl 6007	. . . . 5 |- ( ( A e. P. /\ ( B F C ) e. P. ) -> ( A F ( B F C ) ) e. P. )
14	12, 13	sylan2 284	. . . 4 |- ( ( A e. P. /\ ( B e. P. /\ C e. P. ) ) -> ( A F ( B F C ) ) e. P. )
15	14	3impb 1194	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A F ( B F C ) ) e. P. )
16		preqlu 7434	. . 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 409	. 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 939	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3586   X. cxp 4609   dom cdm 4611   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   P.cnp 7253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-qs 6519  df-ni 7266  df-nqqs 7310  df-inp 7428

This theorem is referenced by:  addassprg  7541  mulassprg  7543