Theorem xrge0iifhom 29983

Description: The defined function from the closed unit interval and the extended nonnegative reals is also a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)

Hypotheses

Ref	Expression
xrge0iifhmeo.1	|- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) )
xrge0iifhmeo.k	|- J = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )

Assertion

Ref	Expression
xrge0iifhom	|- ( ( X e. ( 0 [,] 1 ) /\ Y e. ( 0 [,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )

Distinct variable groups:   x, X   x, F   x, Y

Allowed substitution hint:   J( x)

Proof of Theorem xrge0iifhom

Step	Hyp	Ref	Expression
1		0xr 10086	. . . . . 6 |- 0 e. RR*
2		1re 10039	. . . . . . 7 |- 1 e. RR
3	2	rexri 10097	. . . . . 6 |- 1 e. RR*
4		0le1 10551	. . . . . 6 |- 0 <_ 1
5		snunioc 12300	. . . . . 6 |- ( ( 0 e. RR* /\ 1 e. RR* /\ 0 <_ 1 ) -> ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 ) )
6	1, 3, 4, 5	mp3an 1424	. . . . 5 |- ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 )
7	6	eleq2i 2693	. . . 4 |- ( Y e. ( { 0 } u. ( 0 (,] 1 ) ) <-> Y e. ( 0 [,] 1 ) )
8		elun 3753	. . . 4 |- ( Y e. ( { 0 } u. ( 0 (,] 1 ) ) <-> ( Y e. { 0 } \/ Y e. ( 0 (,] 1 ) ) )
9	7, 8	bitr3i 266	. . 3 |- ( Y e. ( 0 [,] 1 ) <-> ( Y e. { 0 } \/ Y e. ( 0 (,] 1 ) ) )
10		elsni 4194	. . . 4 |- ( Y e. { 0 } -> Y = 0 )
11	10	orim1i 539	. . 3 |- ( ( Y e. { 0 } \/ Y e. ( 0 (,] 1 ) ) -> ( Y = 0 \/ Y e. ( 0 (,] 1 ) ) )
12	9, 11	sylbi 207	. 2 |- ( Y e. ( 0 [,] 1 ) -> ( Y = 0 \/ Y e. ( 0 (,] 1 ) ) )
13		0elunit 12290	. . . . . . . 8 |- 0 e. ( 0 [,] 1 )
14		iftrue 4092	. . . . . . . . 9 |- ( x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) = +oo )
15		xrge0iifhmeo.1	. . . . . . . . 9 |- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) )
16		pnfex 10093	. . . . . . . . 9 |- +oo e. _V
17	14, 15, 16	fvmpt 6282	. . . . . . . 8 |- ( 0 e. ( 0 [,] 1 ) -> ( F ` 0 ) = +oo )
18	13, 17	ax-mp 5	. . . . . . 7 |- ( F ` 0 ) = +oo
19	18	oveq2i 6661	. . . . . 6 |- ( ( F ` X ) +e ( F ` 0 ) ) = ( ( F ` X ) +e +oo )
20		eqeq1 2626	. . . . . . . . . . 11 |- ( x = X -> ( x = 0 <-> X = 0 ) )
21		fveq2 6191	. . . . . . . . . . . 12 |- ( x = X -> ( log ` x ) = ( log ` X ) )
22	21	negeqd 10275	. . . . . . . . . . 11 |- ( x = X -> -u ( log ` x ) = -u ( log ` X ) )
23	20, 22	ifbieq2d 4111	. . . . . . . . . 10 |- ( x = X -> if ( x = 0 , +oo , -u ( log ` x ) ) = if ( X = 0 , +oo , -u ( log ` X ) ) )
24		negex 10279	. . . . . . . . . . 11 |- -u ( log ` X ) e. _V
25	16, 24	ifex 4156	. . . . . . . . . 10 |- if ( X = 0 , +oo , -u ( log ` X ) ) e. _V
26	23, 15, 25	fvmpt 6282	. . . . . . . . 9 |- ( X e. ( 0 [,] 1 ) -> ( F ` X ) = if ( X = 0 , +oo , -u ( log ` X ) ) )
27		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
28	27	a1i 11	. . . . . . . . . 10 |- ( ( X e. ( 0 [,] 1 ) /\ X = 0 ) -> +oo e. RR* )
29		elunitrn 29943	. . . . . . . . . . . . . . 15 |- ( X e. ( 0 [,] 1 ) -> X e. RR )
30	29	adantr 481	. . . . . . . . . . . . . 14 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> X e. RR )
31		elunitge0 29945	. . . . . . . . . . . . . . . 16 |- ( X e. ( 0 [,] 1 ) -> 0 <_ X )
32	31	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> 0 <_ X )
33		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> -. X = 0 )
34	33	neqned 2801	. . . . . . . . . . . . . . 15 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> X =/= 0 )
35	30, 32, 34	ne0gt0d 10174	. . . . . . . . . . . . . 14 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> 0 < X )
36	30, 35	elrpd 11869	. . . . . . . . . . . . 13 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> X e. RR+ )
37	36	relogcld 24369	. . . . . . . . . . . 12 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> ( log ` X ) e. RR )
38	37	renegcld 10457	. . . . . . . . . . 11 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> -u ( log ` X ) e. RR )
39	38	rexrd 10089	. . . . . . . . . 10 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> -u ( log ` X ) e. RR* )
40	28, 39	ifclda 4120	. . . . . . . . 9 |- ( X e. ( 0 [,] 1 ) -> if ( X = 0 , +oo , -u ( log ` X ) ) e. RR* )
41	26, 40	eqeltrd 2701	. . . . . . . 8 |- ( X e. ( 0 [,] 1 ) -> ( F ` X ) e. RR* )
42	41	adantr 481	. . . . . . 7 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( F ` X ) e. RR* )
43		neeq1 2856	. . . . . . . . . 10 |- ( +oo = if ( X = 0 , +oo , -u ( log ` X ) ) -> ( +oo =/= -oo <-> if ( X = 0 , +oo , -u ( log ` X ) ) =/= -oo ) )
44		neeq1 2856	. . . . . . . . . 10 |- ( -u ( log ` X ) = if ( X = 0 , +oo , -u ( log ` X ) ) -> ( -u ( log ` X ) =/= -oo <-> if ( X = 0 , +oo , -u ( log ` X ) ) =/= -oo ) )
45		pnfnemnf 10094	. . . . . . . . . . 11 |- +oo =/= -oo
46	45	a1i 11	. . . . . . . . . 10 |- ( ( X e. ( 0 [,] 1 ) /\ X = 0 ) -> +oo =/= -oo )
47	38	renemnfd 10091	. . . . . . . . . 10 |- ( ( X e. ( 0 [,] 1 ) /\ -. X = 0 ) -> -u ( log ` X ) =/= -oo )
48	43, 44, 46, 47	ifbothda 4123	. . . . . . . . 9 |- ( X e. ( 0 [,] 1 ) -> if ( X = 0 , +oo , -u ( log ` X ) ) =/= -oo )
49	26, 48	eqnetrd 2861	. . . . . . . 8 |- ( X e. ( 0 [,] 1 ) -> ( F ` X ) =/= -oo )
50	49	adantr 481	. . . . . . 7 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( F ` X ) =/= -oo )
51		xaddpnf1 12057	. . . . . . 7 |- ( ( ( F ` X ) e. RR* /\ ( F ` X ) =/= -oo ) -> ( ( F ` X ) +e +oo ) = +oo )
52	42, 50, 51	syl2anc 693	. . . . . 6 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( ( F ` X ) +e +oo ) = +oo )
53	19, 52	syl5eq 2668	. . . . 5 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( ( F ` X ) +e ( F ` 0 ) ) = +oo )
54		unitsscn 29942	. . . . . . . . 9 |- ( 0 [,] 1 ) C_ CC
55		simpl 473	. . . . . . . . 9 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> X e. ( 0 [,] 1 ) )
56	54, 55	sseldi 3601	. . . . . . . 8 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> X e. CC )
57	56	mul01d 10235	. . . . . . 7 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( X x. 0 ) = 0 )
58	57	fveq2d 6195	. . . . . 6 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( F ` ( X x. 0 ) ) = ( F ` 0 ) )
59	58, 18	syl6eq 2672	. . . . 5 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( F ` ( X x. 0 ) ) = +oo )
60	53, 59	eqtr4d 2659	. . . 4 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( ( F ` X ) +e ( F ` 0 ) ) = ( F ` ( X x. 0 ) ) )
61		simpr 477	. . . . . 6 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> Y = 0 )
62	61	fveq2d 6195	. . . . 5 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( F ` Y ) = ( F ` 0 ) )
63	62	oveq2d 6666	. . . 4 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( ( F ` X ) +e ( F ` Y ) ) = ( ( F ` X ) +e ( F ` 0 ) ) )
64	61	oveq2d 6666	. . . . 5 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( X x. Y ) = ( X x. 0 ) )
65	64	fveq2d 6195	. . . 4 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( F ` ( X x. Y ) ) = ( F ` ( X x. 0 ) ) )
66	60, 63, 65	3eqtr4rd 2667	. . 3 |- ( ( X e. ( 0 [,] 1 ) /\ Y = 0 ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )
67	6	eleq2i 2693	. . . . . 6 |- ( X e. ( { 0 } u. ( 0 (,] 1 ) ) <-> X e. ( 0 [,] 1 ) )
68		elun 3753	. . . . . 6 |- ( X e. ( { 0 } u. ( 0 (,] 1 ) ) <-> ( X e. { 0 } \/ X e. ( 0 (,] 1 ) ) )
69	67, 68	bitr3i 266	. . . . 5 |- ( X e. ( 0 [,] 1 ) <-> ( X e. { 0 } \/ X e. ( 0 (,] 1 ) ) )
70		elsni 4194	. . . . . 6 |- ( X e. { 0 } -> X = 0 )
71	70	orim1i 539	. . . . 5 |- ( ( X e. { 0 } \/ X e. ( 0 (,] 1 ) ) -> ( X = 0 \/ X e. ( 0 (,] 1 ) ) )
72	69, 71	sylbi 207	. . . 4 |- ( X e. ( 0 [,] 1 ) -> ( X = 0 \/ X e. ( 0 (,] 1 ) ) )
73	18	oveq1i 6660	. . . . . . . 8 |- ( ( F ` 0 ) +e ( F ` Y ) ) = ( +oo +e ( F ` Y ) )
74		simpr 477	. . . . . . . . . 10 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> Y e. ( 0 (,] 1 ) )
75	15	xrge0iifcv 29980	. . . . . . . . . . . 12 |- ( Y e. ( 0 (,] 1 ) -> ( F ` Y ) = -u ( log ` Y ) )
76		0le0 11110	. . . . . . . . . . . . . . . . 17 |- 0 <_ 0
77		ltpnf 11954	. . . . . . . . . . . . . . . . . 18 |- ( 1 e. RR -> 1 < +oo )
78	2, 77	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- 1 < +oo
79		iocssioo 12263	. . . . . . . . . . . . . . . . 17 |- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 <_ 0 /\ 1 < +oo ) ) -> ( 0 (,] 1 ) C_ ( 0 (,) +oo ) )
80	1, 27, 76, 78, 79	mp4an 709	. . . . . . . . . . . . . . . 16 |- ( 0 (,] 1 ) C_ ( 0 (,) +oo )
81		ioorp 12251	. . . . . . . . . . . . . . . 16 |- ( 0 (,) +oo ) = RR+
82	80, 81	sseqtri 3637	. . . . . . . . . . . . . . 15 |- ( 0 (,] 1 ) C_ RR+
83	82	sseli 3599	. . . . . . . . . . . . . 14 |- ( Y e. ( 0 (,] 1 ) -> Y e. RR+ )
84	83	relogcld 24369	. . . . . . . . . . . . 13 |- ( Y e. ( 0 (,] 1 ) -> ( log ` Y ) e. RR )
85	84	renegcld 10457	. . . . . . . . . . . 12 |- ( Y e. ( 0 (,] 1 ) -> -u ( log ` Y ) e. RR )
86	75, 85	eqeltrd 2701	. . . . . . . . . . 11 |- ( Y e. ( 0 (,] 1 ) -> ( F ` Y ) e. RR )
87	86	rexrd 10089	. . . . . . . . . 10 |- ( Y e. ( 0 (,] 1 ) -> ( F ` Y ) e. RR* )
88	74, 87	syl 17	. . . . . . . . 9 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( F ` Y ) e. RR* )
89	86	renemnfd 10091	. . . . . . . . . 10 |- ( Y e. ( 0 (,] 1 ) -> ( F ` Y ) =/= -oo )
90	74, 89	syl 17	. . . . . . . . 9 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( F ` Y ) =/= -oo )
91		xaddpnf2 12058	. . . . . . . . 9 |- ( ( ( F ` Y ) e. RR* /\ ( F ` Y ) =/= -oo ) -> ( +oo +e ( F ` Y ) ) = +oo )
92	88, 90, 91	syl2anc 693	. . . . . . . 8 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( +oo +e ( F ` Y ) ) = +oo )
93	73, 92	syl5eq 2668	. . . . . . 7 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( ( F ` 0 ) +e ( F ` Y ) ) = +oo )
94		rpssre 11843	. . . . . . . . . . . . 13 |- RR+ C_ RR
95	82, 94	sstri 3612	. . . . . . . . . . . 12 |- ( 0 (,] 1 ) C_ RR
96		ax-resscn 9993	. . . . . . . . . . . 12 |- RR C_ CC
97	95, 96	sstri 3612	. . . . . . . . . . 11 |- ( 0 (,] 1 ) C_ CC
98	97, 74	sseldi 3601	. . . . . . . . . 10 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> Y e. CC )
99	98	mul02d 10234	. . . . . . . . 9 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( 0 x. Y ) = 0 )
100	99	fveq2d 6195	. . . . . . . 8 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( 0 x. Y ) ) = ( F ` 0 ) )
101	100, 18	syl6eq 2672	. . . . . . 7 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( 0 x. Y ) ) = +oo )
102	93, 101	eqtr4d 2659	. . . . . 6 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( ( F ` 0 ) +e ( F ` Y ) ) = ( F ` ( 0 x. Y ) ) )
103		simpl 473	. . . . . . . 8 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> X = 0 )
104	103	fveq2d 6195	. . . . . . 7 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( F ` X ) = ( F ` 0 ) )
105	104	oveq1d 6665	. . . . . 6 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( ( F ` X ) +e ( F ` Y ) ) = ( ( F ` 0 ) +e ( F ` Y ) ) )
106	103	oveq1d 6665	. . . . . . 7 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( X x. Y ) = ( 0 x. Y ) )
107	106	fveq2d 6195	. . . . . 6 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( F ` ( 0 x. Y ) ) )
108	102, 105, 107	3eqtr4rd 2667	. . . . 5 |- ( ( X = 0 /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )
109		simpl 473	. . . . . . . . . 10 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> X e. ( 0 (,] 1 ) )
110	82, 109	sseldi 3601	. . . . . . . . 9 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> X e. RR+ )
111	110	relogcld 24369	. . . . . . . 8 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( log ` X ) e. RR )
112	111	renegcld 10457	. . . . . . 7 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> -u ( log ` X ) e. RR )
113		simpr 477	. . . . . . . . . 10 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> Y e. ( 0 (,] 1 ) )
114	82, 113	sseldi 3601	. . . . . . . . 9 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> Y e. RR+ )
115	114	relogcld 24369	. . . . . . . 8 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( log ` Y ) e. RR )
116	115	renegcld 10457	. . . . . . 7 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> -u ( log ` Y ) e. RR )
117		rexadd 12063	. . . . . . 7 |- ( ( -u ( log ` X ) e. RR /\ -u ( log ` Y ) e. RR ) -> ( -u ( log ` X ) +e -u ( log ` Y ) ) = ( -u ( log ` X ) + -u ( log ` Y ) ) )
118	112, 116, 117	syl2anc 693	. . . . . 6 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( -u ( log ` X ) +e -u ( log ` Y ) ) = ( -u ( log ` X ) + -u ( log ` Y ) ) )
119	15	xrge0iifcv 29980	. . . . . . 7 |- ( X e. ( 0 (,] 1 ) -> ( F ` X ) = -u ( log ` X ) )
120	119, 75	oveqan12d 6669	. . . . . 6 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( ( F ` X ) +e ( F ` Y ) ) = ( -u ( log ` X ) +e -u ( log ` Y ) ) )
121	110	rpred 11872	. . . . . . . . . 10 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> X e. RR )
122	114	rpred 11872	. . . . . . . . . 10 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> Y e. RR )
123	121, 122	remulcld 10070	. . . . . . . . 9 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( X x. Y ) e. RR )
124	110	rpgt0d 11875	. . . . . . . . . 10 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> 0 < X )
125	114	rpgt0d 11875	. . . . . . . . . 10 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> 0 < Y )
126	121, 122, 124, 125	mulgt0d 10192	. . . . . . . . 9 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> 0 < ( X x. Y ) )
127		iocssicc 12261	. . . . . . . . . . . 12 |- ( 0 (,] 1 ) C_ ( 0 [,] 1 )
128	127, 109	sseldi 3601	. . . . . . . . . . 11 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> X e. ( 0 [,] 1 ) )
129	127, 113	sseldi 3601	. . . . . . . . . . 11 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> Y e. ( 0 [,] 1 ) )
130		iimulcl 22736	. . . . . . . . . . 11 |- ( ( X e. ( 0 [,] 1 ) /\ Y e. ( 0 [,] 1 ) ) -> ( X x. Y ) e. ( 0 [,] 1 ) )
131	128, 129, 130	syl2anc 693	. . . . . . . . . 10 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( X x. Y ) e. ( 0 [,] 1 ) )
132		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
133	132, 2	elicc2i 12239	. . . . . . . . . . 11 |- ( ( X x. Y ) e. ( 0 [,] 1 ) <-> ( ( X x. Y ) e. RR /\ 0 <_ ( X x. Y ) /\ ( X x. Y ) <_ 1 ) )
134	133	simp3bi 1078	. . . . . . . . . 10 |- ( ( X x. Y ) e. ( 0 [,] 1 ) -> ( X x. Y ) <_ 1 )
135	131, 134	syl 17	. . . . . . . . 9 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( X x. Y ) <_ 1 )
136		elioc2 12236	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( ( X x. Y ) e. ( 0 (,] 1 ) <-> ( ( X x. Y ) e. RR /\ 0 < ( X x. Y ) /\ ( X x. Y ) <_ 1 ) ) )
137	1, 2, 136	mp2an 708	. . . . . . . . 9 |- ( ( X x. Y ) e. ( 0 (,] 1 ) <-> ( ( X x. Y ) e. RR /\ 0 < ( X x. Y ) /\ ( X x. Y ) <_ 1 ) )
138	123, 126, 135, 137	syl3anbrc 1246	. . . . . . . 8 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( X x. Y ) e. ( 0 (,] 1 ) )
139	15	xrge0iifcv 29980	. . . . . . . 8 |- ( ( X x. Y ) e. ( 0 (,] 1 ) -> ( F ` ( X x. Y ) ) = -u ( log ` ( X x. Y ) ) )
140	138, 139	syl 17	. . . . . . 7 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( X x. Y ) ) = -u ( log ` ( X x. Y ) ) )
141	110, 114	relogmuld 24371	. . . . . . . 8 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( log ` ( X x. Y ) ) = ( ( log ` X ) + ( log ` Y ) ) )
142	141	negeqd 10275	. . . . . . 7 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> -u ( log ` ( X x. Y ) ) = -u ( ( log ` X ) + ( log ` Y ) ) )
143	111	recnd 10068	. . . . . . . 8 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( log ` X ) e. CC )
144	115	recnd 10068	. . . . . . . 8 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( log ` Y ) e. CC )
145	143, 144	negdid 10405	. . . . . . 7 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> -u ( ( log ` X ) + ( log ` Y ) ) = ( -u ( log ` X ) + -u ( log ` Y ) ) )
146	140, 142, 145	3eqtrd 2660	. . . . . 6 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( -u ( log ` X ) + -u ( log ` Y ) ) )
147	118, 120, 146	3eqtr4rd 2667	. . . . 5 |- ( ( X e. ( 0 (,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )
148	108, 147	jaoian 824	. . . 4 |- ( ( ( X = 0 \/ X e. ( 0 (,] 1 ) ) /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )
149	72, 148	sylan 488	. . 3 |- ( ( X e. ( 0 [,] 1 ) /\ Y e. ( 0 (,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )
150	66, 149	jaodan 826	. 2 |- ( ( X e. ( 0 [,] 1 ) /\ ( Y = 0 \/ Y e. ( 0 (,] 1 ) ) ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )
151	12, 150	sylan2 491	1 |- ( ( X e. ( 0 [,] 1 ) /\ Y e. ( 0 [,] 1 ) ) -> ( F ` ( X x. Y ) ) = ( ( F ` X ) +e ( F ` Y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   RR+crp 11832   +ecxad 11944   (,)cioo 12175   (,]cioc 12176   [,]cicc 12178   ↾_t crest 16081  ordTopcordt 16159   logclog 24301

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  ax-addf 10015  ax-mulf 10016

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-iin 4523  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-supp 7296  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-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  xrge0iifmhm  29985  xrge0pluscn  29986