Theorem stoweidlem22 40239

Description: If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem22.8	|- F/ t ph
stoweidlem22.9	|- F/_ t F
stoweidlem22.10	|- F/_ t G
stoweidlem22.1	|- H = ( t e. T |-> ( ( F ` t ) - ( G ` t ) ) )
stoweidlem22.2	|- I = ( t e. T |-> -u 1 )
stoweidlem22.3	|- L = ( t e. T |-> ( ( I ` t ) x. ( G ` t ) ) )
stoweidlem22.4	|- ( ( ph /\ f e. A ) -> f : T --> RR )
stoweidlem22.5	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
stoweidlem22.6	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
stoweidlem22.7	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )

Assertion

Ref	Expression
stoweidlem22	|- ( ( ph /\ F e. A /\ G e. A ) -> ( t e. T |-> ( ( F ` t ) - ( G ` t ) ) ) e. A )

Distinct variable groups:   f, g, t, A   f, F, g   f, G, g   f, I, g   T, f, g, t   ph, f, g   g, L   x, t, A   x, T   ph, x

Allowed substitution hints:   ph( t)   F( x, t)   G( x, t)   H( x, t, f, g)   I( x, t)   L( x, t, f)

Proof of Theorem stoweidlem22

Step	Hyp	Ref	Expression
1		stoweidlem22.8	. . . 4 |- F/ t ph
2		stoweidlem22.9	. . . . 5 |- F/_ t F
3	2	nfel1 2779	. . . 4 |- F/ t F e. A
4		stoweidlem22.10	. . . . 5 |- F/_ t G
5	4	nfel1 2779	. . . 4 |- F/ t G e. A
6	1, 3, 5	nf3an 1831	. . 3 |- F/ t ( ph /\ F e. A /\ G e. A )
7		simpr 477	. . . . . . 7 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> t e. T )
8		simpl1 1064	. . . . . . . . . 10 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ph )
9		stoweidlem22.2	. . . . . . . . . . . 12 |- I = ( t e. T |-> -u 1 )
10		neg1rr 11125	. . . . . . . . . . . . 13 |- -u 1 e. RR
11		stoweidlem22.7	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
12	11	stoweidlem4 40221	. . . . . . . . . . . . 13 |- ( ( ph /\ -u 1 e. RR ) -> ( t e. T |-> -u 1 ) e. A )
13	10, 12	mpan2 707	. . . . . . . . . . . 12 |- ( ph -> ( t e. T |-> -u 1 ) e. A )
14	9, 13	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> I e. A )
15		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( f = I -> ( f e. A <-> I e. A ) )
16	15	anbi2d 740	. . . . . . . . . . . . . 14 |- ( f = I -> ( ( ph /\ f e. A ) <-> ( ph /\ I e. A ) ) )
17		feq1 6026	. . . . . . . . . . . . . 14 |- ( f = I -> ( f : T --> RR <-> I : T --> RR ) )
18	16, 17	imbi12d 334	. . . . . . . . . . . . 13 |- ( f = I -> ( ( ( ph /\ f e. A ) -> f : T --> RR ) <-> ( ( ph /\ I e. A ) -> I : T --> RR ) ) )
19		stoweidlem22.4	. . . . . . . . . . . . 13 |- ( ( ph /\ f e. A ) -> f : T --> RR )
20	18, 19	vtoclg 3266	. . . . . . . . . . . 12 |- ( I e. A -> ( ( ph /\ I e. A ) -> I : T --> RR ) )
21	20	anabsi7 860	. . . . . . . . . . 11 |- ( ( ph /\ I e. A ) -> I : T --> RR )
22	14, 21	mpdan 702	. . . . . . . . . 10 |- ( ph -> I : T --> RR )
23	8, 22	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> I : T --> RR )
24	23, 7	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( I ` t ) e. RR )
25		simpl3 1066	. . . . . . . . 9 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> G e. A )
26		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( f = G -> ( f e. A <-> G e. A ) )
27	26	anbi2d 740	. . . . . . . . . . . . . 14 |- ( f = G -> ( ( ph /\ f e. A ) <-> ( ph /\ G e. A ) ) )
28		feq1 6026	. . . . . . . . . . . . . 14 |- ( f = G -> ( f : T --> RR <-> G : T --> RR ) )
29	27, 28	imbi12d 334	. . . . . . . . . . . . 13 |- ( f = G -> ( ( ( ph /\ f e. A ) -> f : T --> RR ) <-> ( ( ph /\ G e. A ) -> G : T --> RR ) ) )
30	29, 19	vtoclg 3266	. . . . . . . . . . . 12 |- ( G e. A -> ( ( ph /\ G e. A ) -> G : T --> RR ) )
31	30	anabsi7 860	. . . . . . . . . . 11 |- ( ( ph /\ G e. A ) -> G : T --> RR )
32	31	3adant3 1081	. . . . . . . . . 10 |- ( ( ph /\ G e. A /\ t e. T ) -> G : T --> RR )
33		simp3 1063	. . . . . . . . . 10 |- ( ( ph /\ G e. A /\ t e. T ) -> t e. T )
34	32, 33	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ G e. A /\ t e. T ) -> ( G ` t ) e. RR )
35	8, 25, 7, 34	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( G ` t ) e. RR )
36	24, 35	remulcld 10070	. . . . . . 7 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( ( I ` t ) x. ( G ` t ) ) e. RR )
37		stoweidlem22.3	. . . . . . . 8 |- L = ( t e. T |-> ( ( I ` t ) x. ( G ` t ) ) )
38	37	fvmpt2 6291	. . . . . . 7 |- ( ( t e. T /\ ( ( I ` t ) x. ( G ` t ) ) e. RR ) -> ( L ` t ) = ( ( I ` t ) x. ( G ` t ) ) )
39	7, 36, 38	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( L ` t ) = ( ( I ` t ) x. ( G ` t ) ) )
40	9	fvmpt2 6291	. . . . . . . . 9 |- ( ( t e. T /\ -u 1 e. RR ) -> ( I ` t ) = -u 1 )
41	10, 40	mpan2 707	. . . . . . . 8 |- ( t e. T -> ( I ` t ) = -u 1 )
42	41	adantl 482	. . . . . . 7 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( I ` t ) = -u 1 )
43	42	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( ( I ` t ) x. ( G ` t ) ) = ( -u 1 x. ( G ` t ) ) )
44	35	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( G ` t ) e. CC )
45	44	mulm1d 10482	. . . . . 6 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( -u 1 x. ( G ` t ) ) = -u ( G ` t ) )
46	39, 43, 45	3eqtrd 2660	. . . . 5 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( L ` t ) = -u ( G ` t ) )
47	46	oveq2d 6666	. . . 4 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( ( F ` t ) + ( L ` t ) ) = ( ( F ` t ) + -u ( G ` t ) ) )
48		simpl2 1065	. . . . . . . 8 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> F e. A )
49		eleq1 2689	. . . . . . . . . . . 12 |- ( f = F -> ( f e. A <-> F e. A ) )
50	49	anbi2d 740	. . . . . . . . . . 11 |- ( f = F -> ( ( ph /\ f e. A ) <-> ( ph /\ F e. A ) ) )
51		feq1 6026	. . . . . . . . . . 11 |- ( f = F -> ( f : T --> RR <-> F : T --> RR ) )
52	50, 51	imbi12d 334	. . . . . . . . . 10 |- ( f = F -> ( ( ( ph /\ f e. A ) -> f : T --> RR ) <-> ( ( ph /\ F e. A ) -> F : T --> RR ) ) )
53	52, 19	vtoclg 3266	. . . . . . . . 9 |- ( F e. A -> ( ( ph /\ F e. A ) -> F : T --> RR ) )
54	53	anabsi7 860	. . . . . . . 8 |- ( ( ph /\ F e. A ) -> F : T --> RR )
55	8, 48, 54	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> F : T --> RR )
56	55, 7	ffvelrnd 6360	. . . . . 6 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( F ` t ) e. RR )
57	56	recnd 10068	. . . . 5 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( F ` t ) e. CC )
58	57, 44	negsubd 10398	. . . 4 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( ( F ` t ) + -u ( G ` t ) ) = ( ( F ` t ) - ( G ` t ) ) )
59	47, 58	eqtr2d 2657	. . 3 |- ( ( ( ph /\ F e. A /\ G e. A ) /\ t e. T ) -> ( ( F ` t ) - ( G ` t ) ) = ( ( F ` t ) + ( L ` t ) ) )
60	6, 59	mpteq2da 4743	. 2 |- ( ( ph /\ F e. A /\ G e. A ) -> ( t e. T |-> ( ( F ` t ) - ( G ` t ) ) ) = ( t e. T |-> ( ( F ` t ) + ( L ` t ) ) ) )
61	14	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ F e. A /\ G e. A ) -> I e. A )
62		nfmpt1 4747	. . . . . . . 8 |- F/_ t ( t e. T |-> -u 1 )
63	9, 62	nfcxfr 2762	. . . . . . 7 |- F/_ t I
64	63	nfeq2 2780	. . . . . 6 |- F/ t f = I
65	4	nfeq2 2780	. . . . . 6 |- F/ t g = G
66		stoweidlem22.6	. . . . . 6 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
67	64, 65, 66	stoweidlem6 40223	. . . . 5 |- ( ( ph /\ I e. A /\ G e. A ) -> ( t e. T |-> ( ( I ` t ) x. ( G ` t ) ) ) e. A )
68	61, 67	syld3an2 1373	. . . 4 |- ( ( ph /\ F e. A /\ G e. A ) -> ( t e. T |-> ( ( I ` t ) x. ( G ` t ) ) ) e. A )
69	37, 68	syl5eqel 2705	. . 3 |- ( ( ph /\ F e. A /\ G e. A ) -> L e. A )
70		stoweidlem22.5	. . . 4 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
71		nfmpt1 4747	. . . . 5 |- F/_ t ( t e. T |-> ( ( I ` t ) x. ( G ` t ) ) )
72	37, 71	nfcxfr 2762	. . . 4 |- F/_ t L
73	70, 2, 72	stoweidlem8 40225	. . 3 |- ( ( ph /\ F e. A /\ L e. A ) -> ( t e. T |-> ( ( F ` t ) + ( L ` t ) ) ) e. A )
74	69, 73	syld3an3 1371	. 2 |- ( ( ph /\ F e. A /\ G e. A ) -> ( t e. T |-> ( ( F ` t ) + ( L ` t ) ) ) e. A )
75	60, 74	eqeltrd 2701	1 |- ( ( ph /\ F e. A /\ G e. A ) -> ( t e. T |-> ( ( F ` t ) - ( G ` t ) ) ) e. A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269

This theorem is referenced by:  stoweidlem33  40250