Theorem fnpprod 5844

Description: Functionhood law for parallel product. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
fnpprod	|- ((F Fn A /\ G Fn B) -> PProd (F, G) Fn (A X. B))

Proof of Theorem fnpprod

Dummy variables a b c d e f g h x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ee4anv 1915	. . . . . . . . 9 |- (E.aE.bE.eE.f(E.cE.d(x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ E.gE.h(x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) <-> (E.aE.bE.cE.d(x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ E.eE.fE.gE.h(x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))))
2		ee4anv 1915	. . . . . . . . . . 11 |- (E.cE.dE.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) <-> (E.cE.d(x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ E.gE.h(x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))))
3	2	2exbii 1583	. . . . . . . . . 10 |- (E.eE.fE.cE.dE.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) <-> E.eE.f(E.cE.d(x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ E.gE.h(x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))))
4	3	2exbii 1583	. . . . . . . . 9 |- (E.aE.bE.eE.fE.cE.dE.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) <-> E.aE.bE.eE.f(E.cE.d(x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ E.gE.h(x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))))
5		brpprod 5840	. . . . . . . . . 10 |- (x PProd (F, G)y <-> E.aE.bE.cE.d(x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)))
6		brpprod 5840	. . . . . . . . . 10 |- (x PProd (F, G)z <-> E.eE.fE.gE.h(x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)))
7	5, 6	anbi12i 678	. . . . . . . . 9 |- ((x PProd (F, G)y /\ x PProd (F, G)z) <-> (E.aE.bE.cE.d(x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ E.eE.fE.gE.h(x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))))
8	1, 4, 7	3bitr4ri 269	. . . . . . . 8 |- ((x PProd (F, G)y /\ x PProd (F, G)z) <-> E.aE.bE.eE.fE.cE.dE.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))))
9		an42 798	. . . . . . . . . . . . . . . . . . . . . 22 |- (((aFc /\ bGd) /\ (aFg /\ bGh)) <-> ((aFc /\ aFg) /\ (bGh /\ bGd)))
10		fununiq 5518	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((Fun F /\ aFc /\ aFg) -> c = g)
11	10	3expib 1154	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Fun F -> ((aFc /\ aFg) -> c = g))
12		fununiq 5518	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((Fun G /\ bGh /\ bGd) -> h = d)
13	12	eqcomd 2358	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((Fun G /\ bGh /\ bGd) -> d = h)
14	13	3expib 1154	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Fun G -> ((bGh /\ bGd) -> d = h))
15	11, 14	im2anan9 808	. . . . . . . . . . . . . . . . . . . . . 22 |- ((Fun F /\ Fun G) -> (((aFc /\ aFg) /\ (bGh /\ bGd)) -> (c = g /\ d = h)))
16	9, 15	syl5bi 208	. . . . . . . . . . . . . . . . . . . . 21 |- ((Fun F /\ Fun G) -> (((aFc /\ bGd) /\ (aFg /\ bGh)) -> (c = g /\ d = h)))
17	16	exp3acom23 1372	. . . . . . . . . . . . . . . . . . . 20 |- ((Fun F /\ Fun G) -> ((aFg /\ bGh) -> ((aFc /\ bGd) -> (c = g /\ d = h))))
18		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . 23 |- (e = a -> (eFg <-> aFg))
19		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . 23 |- (f = b -> (fGh <-> bGh))
20	18, 19	bi2anan9 843	. . . . . . . . . . . . . . . . . . . . . 22 |- ((e = a /\ f = b) -> ((eFg /\ fGh) <-> (aFg /\ bGh)))
21	20	adantr 451	. . . . . . . . . . . . . . . . . . . . 21 |- (((e = a /\ f = b) /\ z = <.g, h>.) -> ((eFg /\ fGh) <-> (aFg /\ bGh)))
22		eqeq2 2362	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (z = <.g, h>. -> (<.c, d>. = z <-> <.c, d>. = <.g, h>.))
23		opth 4603	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.c, d>. = <.g, h>. <-> (c = g /\ d = h))
24	22, 23	syl6bb 252	. . . . . . . . . . . . . . . . . . . . . . 23 |- (z = <.g, h>. -> (<.c, d>. = z <-> (c = g /\ d = h)))
25	24	imbi2d 307	. . . . . . . . . . . . . . . . . . . . . 22 |- (z = <.g, h>. -> (((aFc /\ bGd) -> <.c, d>. = z) <-> ((aFc /\ bGd) -> (c = g /\ d = h))))
26	25	adantl 452	. . . . . . . . . . . . . . . . . . . . 21 |- (((e = a /\ f = b) /\ z = <.g, h>.) -> (((aFc /\ bGd) -> <.c, d>. = z) <-> ((aFc /\ bGd) -> (c = g /\ d = h))))
27	21, 26	imbi12d 311	. . . . . . . . . . . . . . . . . . . 20 |- (((e = a /\ f = b) /\ z = <.g, h>.) -> (((eFg /\ fGh) -> ((aFc /\ bGd) -> <.c, d>. = z)) <-> ((aFg /\ bGh) -> ((aFc /\ bGd) -> (c = g /\ d = h)))))
28	17, 27	syl5ibrcom 213	. . . . . . . . . . . . . . . . . . 19 |- ((Fun F /\ Fun G) -> (((e = a /\ f = b) /\ z = <.g, h>.) -> ((eFg /\ fGh) -> ((aFc /\ bGd) -> <.c, d>. = z))))
29	28	exp3a 425	. . . . . . . . . . . . . . . . . 18 |- ((Fun F /\ Fun G) -> ((e = a /\ f = b) -> (z = <.g, h>. -> ((eFg /\ fGh) -> ((aFc /\ bGd) -> <.c, d>. = z)))))
30	29	3impd 1165	. . . . . . . . . . . . . . . . 17 |- ((Fun F /\ Fun G) -> (((e = a /\ f = b) /\ z = <.g, h>. /\ (eFg /\ fGh)) -> ((aFc /\ bGd) -> <.c, d>. = z)))
31	30	com23 72	. . . . . . . . . . . . . . . 16 |- ((Fun F /\ Fun G) -> ((aFc /\ bGd) -> (((e = a /\ f = b) /\ z = <.g, h>. /\ (eFg /\ fGh)) -> <.c, d>. = z)))
32		eqeq1 2359	. . . . . . . . . . . . . . . . . . . . 21 |- (x = <.a, b>. -> (x = <.e, f>. <-> <.a, b>. = <.e, f>.))
33		eqcom 2355	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.a, b>. = <.e, f>. <-> <.e, f>. = <.a, b>.)
34		opth 4603	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.e, f>. = <.a, b>. <-> (e = a /\ f = b))
35	33, 34	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 |- (<.a, b>. = <.e, f>. <-> (e = a /\ f = b))
36	32, 35	syl6bb 252	. . . . . . . . . . . . . . . . . . . 20 |- (x = <.a, b>. -> (x = <.e, f>. <-> (e = a /\ f = b)))
37	36	3anbi1d 1256	. . . . . . . . . . . . . . . . . . 19 |- (x = <.a, b>. -> ((x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)) <-> ((e = a /\ f = b) /\ z = <.g, h>. /\ (eFg /\ fGh))))
38	37	adantr 451	. . . . . . . . . . . . . . . . . 18 |- ((x = <.a, b>. /\ y = <.c, d>.) -> ((x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)) <-> ((e = a /\ f = b) /\ z = <.g, h>. /\ (eFg /\ fGh))))
39		eqeq1 2359	. . . . . . . . . . . . . . . . . . 19 |- (y = <.c, d>. -> (y = z <-> <.c, d>. = z))
40	39	adantl 452	. . . . . . . . . . . . . . . . . 18 |- ((x = <.a, b>. /\ y = <.c, d>.) -> (y = z <-> <.c, d>. = z))
41	38, 40	imbi12d 311	. . . . . . . . . . . . . . . . 17 |- ((x = <.a, b>. /\ y = <.c, d>.) -> (((x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)) -> y = z) <-> (((e = a /\ f = b) /\ z = <.g, h>. /\ (eFg /\ fGh)) -> <.c, d>. = z)))
42	41	imbi2d 307	. . . . . . . . . . . . . . . 16 |- ((x = <.a, b>. /\ y = <.c, d>.) -> (((aFc /\ bGd) -> ((x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)) -> y = z)) <-> ((aFc /\ bGd) -> (((e = a /\ f = b) /\ z = <.g, h>. /\ (eFg /\ fGh)) -> <.c, d>. = z))))
43	31, 42	syl5ibrcom 213	. . . . . . . . . . . . . . 15 |- ((Fun F /\ Fun G) -> ((x = <.a, b>. /\ y = <.c, d>.) -> ((aFc /\ bGd) -> ((x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)) -> y = z))))
44	43	exp3a 425	. . . . . . . . . . . . . 14 |- ((Fun F /\ Fun G) -> (x = <.a, b>. -> (y = <.c, d>. -> ((aFc /\ bGd) -> ((x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)) -> y = z)))))
45	44	3impd 1165	. . . . . . . . . . . . 13 |- ((Fun F /\ Fun G) -> ((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) -> ((x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh)) -> y = z)))
46	45	imp3a 420	. . . . . . . . . . . 12 |- ((Fun F /\ Fun G) -> (((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) -> y = z))
47	46	exlimdvv 1637	. . . . . . . . . . 11 |- ((Fun F /\ Fun G) -> (E.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) -> y = z))
48	47	exlimdvv 1637	. . . . . . . . . 10 |- ((Fun F /\ Fun G) -> (E.cE.dE.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) -> y = z))
49	48	exlimdvv 1637	. . . . . . . . 9 |- ((Fun F /\ Fun G) -> (E.eE.fE.cE.dE.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) -> y = z))
50	49	exlimdvv 1637	. . . . . . . 8 |- ((Fun F /\ Fun G) -> (E.aE.bE.eE.fE.cE.dE.gE.h((x = <.a, b>. /\ y = <.c, d>. /\ (aFc /\ bGd)) /\ (x = <.e, f>. /\ z = <.g, h>. /\ (eFg /\ fGh))) -> y = z))
51	8, 50	syl5bi 208	. . . . . . 7 |- ((Fun F /\ Fun G) -> ((x PProd (F, G)y /\ x PProd (F, G)z) -> y = z))
52	51	alrimiv 1631	. . . . . 6 |- ((Fun F /\ Fun G) -> A.z((x PProd (F, G)y /\ x PProd (F, G)z) -> y = z))
53	52	alrimivv 1632	. . . . 5 |- ((Fun F /\ Fun G) -> A.xA.yA.z((x PProd (F, G)y /\ x PProd (F, G)z) -> y = z))
54		dffun2 5120	. . . . 5 |- (Fun PProd (F, G) <-> A.xA.yA.z((x PProd (F, G)y /\ x PProd (F, G)z) -> y = z))
55	53, 54	sylibr 203	. . . 4 |- ((Fun F /\ Fun G) -> Fun PProd (F, G))
56		dmpprod 5841	. . . . 5 |- dom PProd (F, G) = (dom F X. dom G)
57		xpeq12 4804	. . . . 5 |- ((dom F = A /\ dom G = B) -> (dom F X. dom G) = (A X. B))
58	56, 57	syl5eq 2397	. . . 4 |- ((dom F = A /\ dom G = B) -> dom PProd (F, G) = (A X. B))
59	55, 58	anim12i 549	. . 3 |- (((Fun F /\ Fun G) /\ (dom F = A /\ dom G = B)) -> (Fun PProd (F, G) /\ dom PProd (F, G) = (A X. B)))
60	59	an4s 799	. 2 |- (((Fun F /\ dom F = A) /\ (Fun G /\ dom G = B)) -> (Fun PProd (F, G) /\ dom PProd (F, G) = (A X. B)))
61		df-fn 4791	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
62		df-fn 4791	. . 3 |- (G Fn B <-> (Fun G /\ dom G = B))
63	61, 62	anbi12i 678	. 2 |- ((F Fn A /\ G Fn B) <-> ((Fun F /\ dom F = A) /\ (Fun G /\ dom G = B)))
64		df-fn 4791	. 2 |- ( PProd (F, G) Fn (A X. B) <-> (Fun PProd (F, G) /\ dom PProd (F, G) = (A X. B)))
65	60, 63, 64	3imtr4i 257	1 |- ((F Fn A /\ G Fn B) -> PProd (F, G) Fn (A X. B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642  <.cop 4562   class class class wbr 4640   X. cxp 4771  dom cdm 4773  Fun wfun 4776   Fn wfn 4777   PProd cpprod 5738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-2nd 4798  df-txp 5737  df-pprod 5739

This theorem is referenced by:  f1opprod  5845