xrge0iifhom - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0iifhom		Structured version Visualization version GIF version

Theorem xrge0iifhom 31180

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

**Hypotheses**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
xrge0iifhmeo.k	⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))

**Assertion**
Ref	Expression
xrge0iifhom	⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(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 10688	. . . . . 6 ⊢ 0 ∈ ℝ^*
2		1xr 10700	. . . . . 6 ⊢ 1 ∈ ℝ^*
3		0le1 11163	. . . . . 6 ⊢ 0 ≤ 1
4		snunioc 12867	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
5	1, 2, 3, 4	mp3an 1457	. . . . 5 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
6	5	eleq2i 2904	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ 𝑌 ∈ (0[,]1))
7		elun 4125	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
8	6, 7	bitr3i 279	. . 3 ⊢ (𝑌 ∈ (0[,]1) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
9		elsni 4584	. . . 4 ⊢ (𝑌 ∈ {0} → 𝑌 = 0)
10	9	orim1i 906	. . 3 ⊢ ((𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
11	8, 10	sylbi 219	. 2 ⊢ (𝑌 ∈ (0[,]1) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
12		0elunit 12856	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
13		iftrue 4473	. . . . . . . . 9 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
14		xrge0iifhmeo.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
15		pnfex 10694	. . . . . . . . 9 ⊢ +∞ ∈ V
16	13, 14, 15	fvmpt 6768	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
17	12, 16	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) = +∞
18	17	oveq2i 7167	. . . . . 6 ⊢ ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = ((𝐹‘𝑋) +_𝑒 +∞)
19		eqeq1 2825	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0))
20		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋))
21	20	negeqd 10880	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋))
22	19, 21	ifbieq2d 4492	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0, +∞, -(log‘𝑥)) = if(𝑋 = 0, +∞, -(log‘𝑋)))
23		negex 10884	. . . . . . . . . . 11 ⊢ -(log‘𝑋) ∈ V
24	15, 23	ifex 4515	. . . . . . . . . 10 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V
25	22, 14, 24	fvmpt 6768	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)))
26		pnfxr 10695	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ∈ ℝ^*)
28		elunitrn 31140	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ)
29	28	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ)
30		elunitge0 31142	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (0[,]1) → 0 ≤ 𝑋)
31	30	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 ≤ 𝑋)
32		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → ¬ 𝑋 = 0)
33	32	neqned 3023	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ≠ 0)
34	29, 31, 33	ne0gt0d 10777	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 < 𝑋)
35	29, 34	elrpd 12429	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ⁺)
36	35	relogcld 25206	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → (log‘𝑋) ∈ ℝ)
37	36	renegcld 11067	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ)
38	37	rexrd 10691	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ^*)
39	27, 38	ifclda 4501	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ ℝ^*)
40	25, 39	eqeltrd 2913	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ∈ ℝ^*)
41	40	adantr 483	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ∈ ℝ^*)
42		neeq1 3078	. . . . . . . . . 10 ⊢ (+∞ = if(𝑋 = 0, +∞, -(log‘𝑋)) → (+∞ ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
43		neeq1 3078	. . . . . . . . . 10 ⊢ (-(log‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)) → (-(log‘𝑋) ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
44		pnfnemnf 10696	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ≠ -∞)
46	37	renemnfd 10693	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ≠ -∞)
47	42, 43, 45, 46	ifbothda 4504	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞)
48	25, 47	eqnetrd 3083	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ≠ -∞)
49	48	adantr 483	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ≠ -∞)
50		xaddpnf1 12620	. . . . . . 7 ⊢ (((𝐹‘𝑋) ∈ ℝ^* ∧ (𝐹‘𝑋) ≠ -∞) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
51	41, 49, 50	syl2anc 586	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
52	18, 51	syl5eq 2868	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = +∞)
53		unitsscn 31139	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℂ
54		simpl 485	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ (0[,]1))
55	53, 54	sseldi 3965	. . . . . . . 8 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ ℂ)
56	55	mul01d 10839	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 0) = 0)
57	56	fveq2d 6674	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = (𝐹‘0))
58	57, 17	syl6eq 2872	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = +∞)
59	52, 58	eqtr4d 2859	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = (𝐹‘(𝑋 · 0)))
60		simpr 487	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑌 = 0)
61	60	fveq2d 6674	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑌) = (𝐹‘0))
62	61	oveq2d 7172	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘0)))
63	60	oveq2d 7172	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 𝑌) = (𝑋 · 0))
64	63	fveq2d 6674	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(𝑋 · 0)))
65	59, 62, 64	3eqtr4rd 2867	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
66	5	eleq2i 2904	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ 𝑋 ∈ (0[,]1))
67		elun 4125	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
68	66, 67	bitr3i 279	. . . . 5 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
69		elsni 4584	. . . . . 6 ⊢ (𝑋 ∈ {0} → 𝑋 = 0)
70	69	orim1i 906	. . . . 5 ⊢ ((𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
71	68, 70	sylbi 219	. . . 4 ⊢ (𝑋 ∈ (0[,]1) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
72	17	oveq1i 7166	. . . . . . . 8 ⊢ ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (+∞ +_𝑒 (𝐹‘𝑌))
73		simpr 487	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
74	14	xrge0iifcv 31177	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) = -(log‘𝑌))
75		0le0 11739	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
76		1re 10641	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
77		ltpnf 12516	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → 1 < +∞)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ 1 < +∞
79		iocssioo 12828	. . . . . . . . . . . . . . . . 17 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
80	1, 26, 75, 78, 79	mp4an 691	. . . . . . . . . . . . . . . 16 ⊢ (0(,]1) ⊆ (0(,)+∞)
81		ioorp 12815	. . . . . . . . . . . . . . . 16 ⊢ (0(,)+∞) = ℝ⁺
82	80, 81	sseqtri 4003	. . . . . . . . . . . . . . 15 ⊢ (0(,]1) ⊆ ℝ⁺
83	82	sseli 3963	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0(,]1) → 𝑌 ∈ ℝ⁺)
84	83	relogcld 25206	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0(,]1) → (log‘𝑌) ∈ ℝ)
85	84	renegcld 11067	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → -(log‘𝑌) ∈ ℝ)
86	74, 85	eqeltrd 2913	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ)
87	86	rexrd 10691	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ^*)
88	73, 87	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ∈ ℝ^*)
89	86	renemnfd 10693	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ≠ -∞)
90	73, 89	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ≠ -∞)
91		xaddpnf2 12621	. . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ℝ^* ∧ (𝐹‘𝑌) ≠ -∞) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
92	88, 90, 91	syl2anc 586	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
93	72, 92	syl5eq 2868	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = +∞)
94		rpssre 12397	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
95	82, 94	sstri 3976	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ
96		ax-resscn 10594	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
97	95, 96	sstri 3976	. . . . . . . . . . 11 ⊢ (0(,]1) ⊆ ℂ
98	97, 73	sseldi 3965	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℂ)
99	98	mul02d 10838	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (0 · 𝑌) = 0)
100	99	fveq2d 6674	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = (𝐹‘0))
101	100, 17	syl6eq 2872	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = +∞)
102	93, 101	eqtr4d 2859	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (𝐹‘(0 · 𝑌)))
103		simpl 485	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑋 = 0)
104	103	fveq2d 6674	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑋) = (𝐹‘0))
105	104	oveq1d 7171	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘0) +_𝑒 (𝐹‘𝑌)))
106	103	fvoveq1d 7178	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(0 · 𝑌)))
107	102, 105, 106	3eqtr4rd 2867	. . . . 5 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
108		simpl 485	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0(,]1))
109	82, 108	sseldi 3965	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ⁺)
110	109	relogcld 25206	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℝ)
111	110	renegcld 11067	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑋) ∈ ℝ)
112		simpr 487	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
113	82, 112	sseldi 3965	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ⁺)
114	113	relogcld 25206	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℝ)
115	114	renegcld 11067	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑌) ∈ ℝ)
116		rexadd 12626	. . . . . . 7 ⊢ ((-(log‘𝑋) ∈ ℝ ∧ -(log‘𝑌) ∈ ℝ) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
117	111, 115, 116	syl2anc 586	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
118	14	xrge0iifcv 31177	. . . . . . 7 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋))
119	118, 74	oveqan12d 7175	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = (-(log‘𝑋) +_𝑒 -(log‘𝑌)))
120	109	rpred 12432	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ)
121	113	rpred 12432	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ)
122	120, 121	remulcld 10671	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ ℝ)
123	109	rpgt0d 12435	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑋)
124	113	rpgt0d 12435	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑌)
125	120, 121, 123, 124	mulgt0d 10795	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < (𝑋 · 𝑌))
126		iocssicc 12826	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ (0[,]1)
127	126, 108	sseldi 3965	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0[,]1))
128	126, 112	sseldi 3965	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0[,]1))
129		iimulcl 23541	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
130	127, 128, 129	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
131		elicc01 12855	. . . . . . . . . . 11 ⊢ ((𝑋 · 𝑌) ∈ (0[,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 ≤ (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
132	131	simp3bi 1143	. . . . . . . . . 10 ⊢ ((𝑋 · 𝑌) ∈ (0[,]1) → (𝑋 · 𝑌) ≤ 1)
133	130, 132	syl 17	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ≤ 1)
134		elioc2 12800	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1)))
135	1, 76, 134	mp2an 690	. . . . . . . . 9 ⊢ ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
136	122, 125, 133, 135	syl3anbrc 1339	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0(,]1))
137	14	xrge0iifcv 31177	. . . . . . . 8 ⊢ ((𝑋 · 𝑌) ∈ (0(,]1) → (𝐹‘(𝑋 · 𝑌)) = -(log‘(𝑋 · 𝑌)))
138	136, 137	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = -(log‘(𝑋 · 𝑌)))
139	109, 113	relogmuld 25208	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘(𝑋 · 𝑌)) = ((log‘𝑋) + (log‘𝑌)))
140	139	negeqd 10880	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘(𝑋 · 𝑌)) = -((log‘𝑋) + (log‘𝑌)))
141	110	recnd 10669	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℂ)
142	114	recnd 10669	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℂ)
143	141, 142	negdid 11010	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -((log‘𝑋) + (log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
144	138, 140, 143	3eqtrd 2860	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
145	117, 119, 144	3eqtr4rd 2867	. . . . 5 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
146	107, 145	jaoian 953	. . . 4 ⊢ (((𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
147	71, 146	sylan 582	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
148	65, 147	jaodan 954	. 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1))) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
149	11, 148	sylan2 594	1 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∪ cun 3934 ⊆ wss 3936 ifcif 4467 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 -cneg 10871 ℝ⁺crp 12390 +_𝑒 cxad 12506 (,)cioo 12739 (,]cioc 12740 [,]cicc 12742 ↾_t crest 16694 ordTopcordt 16772 logclog 25138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140

This theorem is referenced by: xrge0iifmhm 31182 xrge0pluscn 31183

W3C validator