Theorem ftc1anclem3 33487

Description: Lemma for ftc1anc 33493- the absolute value of the sum of a simple function and _i times another simple function is itself a simple function. (Contributed by Brendan Leahy, 27-May-2018.)

Assertion

Ref	Expression
ftc1anclem3	|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( abs o. ( F oF + ( ( RR X. { _i } ) oF x. G ) ) ) e. dom S.1 )

Proof of Theorem ftc1anclem3

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1ff 23443	. . . . . . . 8 |- ( F e. dom S.1 -> F : RR --> RR )
2	1	ffvelrnda 6359	. . . . . . 7 |- ( ( F e. dom S.1 /\ x e. RR ) -> ( F ` x ) e. RR )
3		i1ff 23443	. . . . . . . 8 |- ( G e. dom S.1 -> G : RR --> RR )
4	3	ffvelrnda 6359	. . . . . . 7 |- ( ( G e. dom S.1 /\ x e. RR ) -> ( G ` x ) e. RR )
5		absreim 14033	. . . . . . 7 |- ( ( ( F ` x ) e. RR /\ ( G ` x ) e. RR ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqr ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) )
6	2, 4, 5	syl2an 494	. . . . . 6 |- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqr ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) )
7	6	anandirs 874	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqr ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) )
8	2	recnd 10068	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ x e. RR ) -> ( F ` x ) e. CC )
9	8	sqvald 13005	. . . . . . . 8 |- ( ( F e. dom S.1 /\ x e. RR ) -> ( ( F ` x ) ^ 2 ) = ( ( F ` x ) x. ( F ` x ) ) )
10	4	recnd 10068	. . . . . . . . 9 |- ( ( G e. dom S.1 /\ x e. RR ) -> ( G ` x ) e. CC )
11	10	sqvald 13005	. . . . . . . 8 |- ( ( G e. dom S.1 /\ x e. RR ) -> ( ( G ` x ) ^ 2 ) = ( ( G ` x ) x. ( G ` x ) ) )
12	9, 11	oveqan12d 6669	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) = ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) )
13	12	anandirs 874	. . . . . 6 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) = ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) )
14	13	fveq2d 6195	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( sqr ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) = ( sqr ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) )
15	7, 14	eqtrd 2656	. . . 4 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqr ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) )
16	15	mpteq2dva 4744	. . 3 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( x e. RR |-> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) ) = ( x e. RR |-> ( sqr ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) )
17		ax-icn 9995	. . . . . . 7 |- _i e. CC
18		mulcl 10020	. . . . . . 7 |- ( ( _i e. CC /\ ( G ` x ) e. CC ) -> ( _i x. ( G ` x ) ) e. CC )
19	17, 10, 18	sylancr 695	. . . . . 6 |- ( ( G e. dom S.1 /\ x e. RR ) -> ( _i x. ( G ` x ) ) e. CC )
20		addcl 10018	. . . . . 6 |- ( ( ( F ` x ) e. CC /\ ( _i x. ( G ` x ) ) e. CC ) -> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) e. CC )
21	8, 19, 20	syl2an 494	. . . . 5 |- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) e. CC )
22	21	anandirs 874	. . . 4 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) e. CC )
23		reex 10027	. . . . . 6 |- RR e. _V
24	23	a1i 11	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> RR e. _V )
25	2	adantlr 751	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( F ` x ) e. RR )
26		ovexd 6680	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( _i x. ( G ` x ) ) e. _V )
27	1	feqmptd 6249	. . . . . 6 |- ( F e. dom S.1 -> F = ( x e. RR |-> ( F ` x ) ) )
28	27	adantr 481	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> F = ( x e. RR |-> ( F ` x ) ) )
29	23	a1i 11	. . . . . . 7 |- ( G e. dom S.1 -> RR e. _V )
30	17	a1i 11	. . . . . . 7 |- ( ( G e. dom S.1 /\ x e. RR ) -> _i e. CC )
31		fconstmpt 5163	. . . . . . . 8 |- ( RR X. { _i } ) = ( x e. RR |-> _i )
32	31	a1i 11	. . . . . . 7 |- ( G e. dom S.1 -> ( RR X. { _i } ) = ( x e. RR |-> _i ) )
33	3	feqmptd 6249	. . . . . . 7 |- ( G e. dom S.1 -> G = ( x e. RR |-> ( G ` x ) ) )
34	29, 30, 4, 32, 33	offval2 6914	. . . . . 6 |- ( G e. dom S.1 -> ( ( RR X. { _i } ) oF x. G ) = ( x e. RR |-> ( _i x. ( G ` x ) ) ) )
35	34	adantl 482	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( RR X. { _i } ) oF x. G ) = ( x e. RR |-> ( _i x. ( G ` x ) ) ) )
36	24, 25, 26, 28, 35	offval2 6914	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF + ( ( RR X. { _i } ) oF x. G ) ) = ( x e. RR |-> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) )
37		absf 14077	. . . . . 6 |- abs : CC --> RR
38	37	a1i 11	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> abs : CC --> RR )
39	38	feqmptd 6249	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> abs = ( y e. CC |-> ( abs ` y ) ) )
40		fveq2 6191	. . . 4 |- ( y = ( ( F ` x ) + ( _i x. ( G ` x ) ) ) -> ( abs ` y ) = ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) )
41	22, 36, 39, 40	fmptco 6396	. . 3 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( abs o. ( F oF + ( ( RR X. { _i } ) oF x. G ) ) ) = ( x e. RR |-> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) ) )
42	8, 8	mulcld 10060	. . . . . 6 |- ( ( F e. dom S.1 /\ x e. RR ) -> ( ( F ` x ) x. ( F ` x ) ) e. CC )
43	10, 10	mulcld 10060	. . . . . 6 |- ( ( G e. dom S.1 /\ x e. RR ) -> ( ( G ` x ) x. ( G ` x ) ) e. CC )
44		addcl 10018	. . . . . 6 |- ( ( ( ( F ` x ) x. ( F ` x ) ) e. CC /\ ( ( G ` x ) x. ( G ` x ) ) e. CC ) -> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) e. CC )
45	42, 43, 44	syl2an 494	. . . . 5 |- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) e. CC )
46	45	anandirs 874	. . . 4 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) e. CC )
47	42	adantlr 751	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( F ` x ) x. ( F ` x ) ) e. CC )
48	43	adantll 750	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( G ` x ) x. ( G ` x ) ) e. CC )
49	23	a1i 11	. . . . . . 7 |- ( F e. dom S.1 -> RR e. _V )
50	49, 2, 2, 27, 27	offval2 6914	. . . . . 6 |- ( F e. dom S.1 -> ( F oF x. F ) = ( x e. RR |-> ( ( F ` x ) x. ( F ` x ) ) ) )
51	50	adantr 481	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) = ( x e. RR |-> ( ( F ` x ) x. ( F ` x ) ) ) )
52	29, 4, 4, 33, 33	offval2 6914	. . . . . 6 |- ( G e. dom S.1 -> ( G oF x. G ) = ( x e. RR |-> ( ( G ` x ) x. ( G ` x ) ) ) )
53	52	adantl 482	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) = ( x e. RR |-> ( ( G ` x ) x. ( G ` x ) ) ) )
54	24, 47, 48, 51, 53	offval2 6914	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) = ( x e. RR |-> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) )
55		sqrtf 14103	. . . . . 6 |- sqr : CC --> CC
56	55	a1i 11	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> sqr : CC --> CC )
57	56	feqmptd 6249	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> sqr = ( y e. CC |-> ( sqr ` y ) ) )
58		fveq2 6191	. . . 4 |- ( y = ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) -> ( sqr ` y ) = ( sqr ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) )
59	46, 54, 57, 58	fmptco 6396	. . 3 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) = ( x e. RR |-> ( sqr ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) )
60	16, 41, 59	3eqtr4d 2666	. 2 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( abs o. ( F oF + ( ( RR X. { _i } ) oF x. G ) ) ) = ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) )
61		elrege0 12278	. . . . . . 7 |- ( x e. ( 0 [,) +oo ) <-> ( x e. RR /\ 0 <_ x ) )
62		resqrtcl 13994	. . . . . . 7 |- ( ( x e. RR /\ 0 <_ x ) -> ( sqr ` x ) e. RR )
63	61, 62	sylbi 207	. . . . . 6 |- ( x e. ( 0 [,) +oo ) -> ( sqr ` x ) e. RR )
64	63	adantl 482	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( 0 [,) +oo ) ) -> ( sqr ` x ) e. RR )
65		id 22	. . . . . . . . 9 |- ( sqr : CC --> CC -> sqr : CC --> CC )
66	65	feqmptd 6249	. . . . . . . 8 |- ( sqr : CC --> CC -> sqr = ( x e. CC |-> ( sqr ` x ) ) )
67	55, 66	ax-mp 5	. . . . . . 7 |- sqr = ( x e. CC |-> ( sqr ` x ) )
68	67	reseq1i 5392	. . . . . 6 |- ( sqr |` ( 0 [,) +oo ) ) = ( ( x e. CC |-> ( sqr ` x ) ) |` ( 0 [,) +oo ) )
69		rge0ssre 12280	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
70		ax-resscn 9993	. . . . . . . 8 |- RR C_ CC
71	69, 70	sstri 3612	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
72		resmpt 5449	. . . . . . 7 |- ( ( 0 [,) +oo ) C_ CC -> ( ( x e. CC |-> ( sqr ` x ) ) |` ( 0 [,) +oo ) ) = ( x e. ( 0 [,) +oo ) |-> ( sqr ` x ) ) )
73	71, 72	ax-mp 5	. . . . . 6 |- ( ( x e. CC |-> ( sqr ` x ) ) |` ( 0 [,) +oo ) ) = ( x e. ( 0 [,) +oo ) |-> ( sqr ` x ) )
74	68, 73	eqtri 2644	. . . . 5 |- ( sqr |` ( 0 [,) +oo ) ) = ( x e. ( 0 [,) +oo ) |-> ( sqr ` x ) )
75	64, 74	fmptd 6385	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqr |` ( 0 [,) +oo ) ) : ( 0 [,) +oo ) --> RR )
76		ge0addcl 12284	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x + y ) e. ( 0 [,) +oo ) )
77	76	adantl 482	. . . . 5 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) ) -> ( x + y ) e. ( 0 [,) +oo ) )
78		oveq12 6659	. . . . . . . . 9 |- ( ( z = F /\ z = F ) -> ( z oF x. z ) = ( F oF x. F ) )
79	78	anidms 677	. . . . . . . 8 |- ( z = F -> ( z oF x. z ) = ( F oF x. F ) )
80	79	feq1d 6030	. . . . . . 7 |- ( z = F -> ( ( z oF x. z ) : RR --> ( 0 [,) +oo ) <-> ( F oF x. F ) : RR --> ( 0 [,) +oo ) ) )
81		i1ff 23443	. . . . . . . . . . . 12 |- ( z e. dom S.1 -> z : RR --> RR )
82	81	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( z e. dom S.1 /\ x e. RR ) -> ( z ` x ) e. RR )
83	82, 82	remulcld 10070	. . . . . . . . . 10 |- ( ( z e. dom S.1 /\ x e. RR ) -> ( ( z ` x ) x. ( z ` x ) ) e. RR )
84	82	msqge0d 10596	. . . . . . . . . 10 |- ( ( z e. dom S.1 /\ x e. RR ) -> 0 <_ ( ( z ` x ) x. ( z ` x ) ) )
85		elrege0 12278	. . . . . . . . . 10 |- ( ( ( z ` x ) x. ( z ` x ) ) e. ( 0 [,) +oo ) <-> ( ( ( z ` x ) x. ( z ` x ) ) e. RR /\ 0 <_ ( ( z ` x ) x. ( z ` x ) ) ) )
86	83, 84, 85	sylanbrc 698	. . . . . . . . 9 |- ( ( z e. dom S.1 /\ x e. RR ) -> ( ( z ` x ) x. ( z ` x ) ) e. ( 0 [,) +oo ) )
87		eqid 2622	. . . . . . . . 9 |- ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) ) = ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) )
88	86, 87	fmptd 6385	. . . . . . . 8 |- ( z e. dom S.1 -> ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) ) : RR --> ( 0 [,) +oo ) )
89	23	a1i 11	. . . . . . . . . 10 |- ( z e. dom S.1 -> RR e. _V )
90	81	feqmptd 6249	. . . . . . . . . 10 |- ( z e. dom S.1 -> z = ( x e. RR |-> ( z ` x ) ) )
91	89, 82, 82, 90, 90	offval2 6914	. . . . . . . . 9 |- ( z e. dom S.1 -> ( z oF x. z ) = ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) ) )
92	91	feq1d 6030	. . . . . . . 8 |- ( z e. dom S.1 -> ( ( z oF x. z ) : RR --> ( 0 [,) +oo ) <-> ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) ) : RR --> ( 0 [,) +oo ) ) )
93	88, 92	mpbird 247	. . . . . . 7 |- ( z e. dom S.1 -> ( z oF x. z ) : RR --> ( 0 [,) +oo ) )
94	80, 93	vtoclga 3272	. . . . . 6 |- ( F e. dom S.1 -> ( F oF x. F ) : RR --> ( 0 [,) +oo ) )
95	94	adantr 481	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) : RR --> ( 0 [,) +oo ) )
96		oveq12 6659	. . . . . . . . 9 |- ( ( z = G /\ z = G ) -> ( z oF x. z ) = ( G oF x. G ) )
97	96	anidms 677	. . . . . . . 8 |- ( z = G -> ( z oF x. z ) = ( G oF x. G ) )
98	97	feq1d 6030	. . . . . . 7 |- ( z = G -> ( ( z oF x. z ) : RR --> ( 0 [,) +oo ) <-> ( G oF x. G ) : RR --> ( 0 [,) +oo ) ) )
99	98, 93	vtoclga 3272	. . . . . 6 |- ( G e. dom S.1 -> ( G oF x. G ) : RR --> ( 0 [,) +oo ) )
100	99	adantl 482	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) : RR --> ( 0 [,) +oo ) )
101		inidm 3822	. . . . 5 |- ( RR i^i RR ) = RR
102	77, 95, 100, 24, 24, 101	off 6912	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) : RR --> ( 0 [,) +oo ) )
103		fco2 6059	. . . 4 |- ( ( ( sqr |` ( 0 [,) +oo ) ) : ( 0 [,) +oo ) --> RR /\ ( ( F oF x. F ) oF + ( G oF x. G ) ) : RR --> ( 0 [,) +oo ) ) -> ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) : RR --> RR )
104	75, 102, 103	syl2anc 693	. . 3 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) : RR --> RR )
105		rnco 5641	. . . 4 |- ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) = ran ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) )
106		ffn 6045	. . . . . . . 8 |- ( sqr : CC --> CC -> sqr Fn CC )
107	55, 106	ax-mp 5	. . . . . . 7 |- sqr Fn CC
108		readdcl 10019	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
109	108	adantl 482	. . . . . . . . . 10 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
110		remulcl 10021	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
111	110	adantl 482	. . . . . . . . . . . 12 |- ( ( F e. dom S.1 /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
112	111, 1, 1, 49, 49, 101	off 6912	. . . . . . . . . . 11 |- ( F e. dom S.1 -> ( F oF x. F ) : RR --> RR )
113	112	adantr 481	. . . . . . . . . 10 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) : RR --> RR )
114	110	adantl 482	. . . . . . . . . . . 12 |- ( ( G e. dom S.1 /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
115	114, 3, 3, 29, 29, 101	off 6912	. . . . . . . . . . 11 |- ( G e. dom S.1 -> ( G oF x. G ) : RR --> RR )
116	115	adantl 482	. . . . . . . . . 10 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) : RR --> RR )
117	109, 113, 116, 24, 24, 101	off 6912	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) : RR --> RR )
118		frn 6053	. . . . . . . . 9 |- ( ( ( F oF x. F ) oF + ( G oF x. G ) ) : RR --> RR -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) C_ RR )
119	117, 118	syl 17	. . . . . . . 8 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) C_ RR )
120	119, 70	syl6ss 3615	. . . . . . 7 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) C_ CC )
121		fnssres 6004	. . . . . . 7 |- ( ( sqr Fn CC /\ ran ( ( F oF x. F ) oF + ( G oF x. G ) ) C_ CC ) -> ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) Fn ran ( ( F oF x. F ) oF + ( G oF x. G ) ) )
122	107, 120, 121	sylancr 695	. . . . . 6 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) Fn ran ( ( F oF x. F ) oF + ( G oF x. G ) ) )
123		id 22	. . . . . . . . . 10 |- ( F e. dom S.1 -> F e. dom S.1 )
124	123, 123	i1fmul 23463	. . . . . . . . 9 |- ( F e. dom S.1 -> ( F oF x. F ) e. dom S.1 )
125	124	adantr 481	. . . . . . . 8 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) e. dom S.1 )
126		id 22	. . . . . . . . . 10 |- ( G e. dom S.1 -> G e. dom S.1 )
127	126, 126	i1fmul 23463	. . . . . . . . 9 |- ( G e. dom S.1 -> ( G oF x. G ) e. dom S.1 )
128	127	adantl 482	. . . . . . . 8 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) e. dom S.1 )
129	125, 128	i1fadd 23462	. . . . . . 7 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 )
130		i1frn 23444	. . . . . . 7 |- ( ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) e. Fin )
131	129, 130	syl 17	. . . . . 6 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) e. Fin )
132		fnfi 8238	. . . . . 6 |- ( ( ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) Fn ran ( ( F oF x. F ) oF + ( G oF x. G ) ) /\ ran ( ( F oF x. F ) oF + ( G oF x. G ) ) e. Fin ) -> ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin )
133	122, 131, 132	syl2anc 693	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin )
134		rnfi 8249	. . . . 5 |- ( ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin -> ran ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin )
135	133, 134	syl 17	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( sqr |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin )
136	105, 135	syl5eqel 2705	. . 3 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin )
137		cnvco 5308	. . . . . . 7 |- `' ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) = ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) o. `' sqr )
138	137	imaeq1i 5463	. . . . . 6 |- ( `' ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) = ( ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) o. `' sqr ) " { x } )
139		imaco 5640	. . . . . 6 |- ( ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) o. `' sqr ) " { x } ) = ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqr " { x } ) )
140	138, 139	eqtri 2644	. . . . 5 |- ( `' ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) = ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqr " { x } ) )
141		i1fima 23445	. . . . . 6 |- ( ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 -> ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqr " { x } ) ) e. dom vol )
142	129, 141	syl 17	. . . . 5 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqr " { x } ) ) e. dom vol )
143	140, 142	syl5eqel 2705	. . . 4 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( `' ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) e. dom vol )
144	143	adantr 481	. . 3 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) ) -> ( `' ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) e. dom vol )
145	140	fveq2i 6194	. . . 4 |- ( vol ` ( `' ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) ) = ( vol ` ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqr " { x } ) ) )
146		eldifsni 4320	. . . . . . . 8 |- ( x e. ( ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> x =/= 0 )
147		c0ex 10034	. . . . . . . . . . . 12 |- 0 e. _V
148	147	elsn 4192	. . . . . . . . . . 11 |- ( 0 e. { x } <-> 0 = x )
149		eqcom 2629	. . . . . . . . . . 11 |- ( 0 = x <-> x = 0 )
150	148, 149	bitri 264	. . . . . . . . . 10 |- ( 0 e. { x } <-> x = 0 )
151	150	necon3bbii 2841	. . . . . . . . 9 |- ( -. 0 e. { x } <-> x =/= 0 )
152		sqrt0 13982	. . . . . . . . . 10 |- ( sqr ` 0 ) = 0
153	152	eleq1i 2692	. . . . . . . . 9 |- ( ( sqr ` 0 ) e. { x } <-> 0 e. { x } )
154	151, 153	xchnxbir 323	. . . . . . . 8 |- ( -. ( sqr ` 0 ) e. { x } <-> x =/= 0 )
155	146, 154	sylibr 224	. . . . . . 7 |- ( x e. ( ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> -. ( sqr ` 0 ) e. { x } )
156	155	olcd 408	. . . . . 6 |- ( x e. ( ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> ( -. 0 e. CC \/ -. ( sqr ` 0 ) e. { x } ) )
157		ianor 509	. . . . . . 7 |- ( -. ( 0 e. CC /\ ( sqr ` 0 ) e. { x } ) <-> ( -. 0 e. CC \/ -. ( sqr ` 0 ) e. { x } ) )
158		elpreima 6337	. . . . . . . 8 |- ( sqr Fn CC -> ( 0 e. ( `' sqr " { x } ) <-> ( 0 e. CC /\ ( sqr ` 0 ) e. { x } ) ) )
159	55, 106, 158	mp2b 10	. . . . . . 7 |- ( 0 e. ( `' sqr " { x } ) <-> ( 0 e. CC /\ ( sqr ` 0 ) e. { x } ) )
160	157, 159	xchnxbir 323	. . . . . 6 |- ( -. 0 e. ( `' sqr " { x } ) <-> ( -. 0 e. CC \/ -. ( sqr ` 0 ) e. { x } ) )
161	156, 160	sylibr 224	. . . . 5 |- ( x e. ( ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> -. 0 e. ( `' sqr " { x } ) )
162		i1fima2 23446	. . . . 5 |- ( ( ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 /\ -. 0 e. ( `' sqr " { x } ) ) -> ( vol ` ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqr " { x } ) ) ) e. RR )
163	129, 161, 162	syl2an 494	. . . 4 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) ) -> ( vol ` ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqr " { x } ) ) ) e. RR )
164	145, 163	syl5eqel 2705	. . 3 |- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( ran ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) ) -> ( vol ` ( `' ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) ) e. RR )
165	104, 136, 144, 164	i1fd 23448	. 2 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqr o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. dom S.1 )
166	60, 165	eqeltrd 2701	1 |- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( abs o. ( F oF + ( ( RR X. { _i } ) oF x. G ) ) ) e. dom S.1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   _ici 9938   + caddc 9939   x. cmul 9941   +oocpnf 10071   <_ cle 10075   2c2 11070   [,)cico 12177   ^cexp 12860   sqrcsqrt 13973   abscabs 13974   volcvol 23232   S.1citg1 23384

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  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  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389

This theorem is referenced by:  ftc1anclem7  33491  ftc1anclem8  33492