xrge0iifhom - Mathbox for Thierry Arnoux

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Theorem xrge0iifhom 33936

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 11308	. . . . . 6 ⊢ 0 ∈ ℝ^*
2		1xr 11320	. . . . . 6 ⊢ 1 ∈ ℝ^*
3		0le1 11786	. . . . . 6 ⊢ 0 ≤ 1
4		snunioc 13520	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
5	1, 2, 3, 4	mp3an 1463	. . . . 5 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
6	5	eleq2i 2833	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ 𝑌 ∈ (0[,]1))
7		elun 4153	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
8	6, 7	bitr3i 277	. . 3 ⊢ (𝑌 ∈ (0[,]1) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
9		elsni 4643	. . . 4 ⊢ (𝑌 ∈ {0} → 𝑌 = 0)
10	9	orim1i 910	. . 3 ⊢ ((𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
11	8, 10	sylbi 217	. 2 ⊢ (𝑌 ∈ (0[,]1) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
12		0elunit 13509	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
13		iftrue 4531	. . . . . . . . 9 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
14		xrge0iifhmeo.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
15		pnfex 11314	. . . . . . . . 9 ⊢ +∞ ∈ V
16	13, 14, 15	fvmpt 7016	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
17	12, 16	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) = +∞
18	17	oveq2i 7442	. . . . . 6 ⊢ ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = ((𝐹‘𝑋) +_𝑒 +∞)
19		eqeq1 2741	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0))
20		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋))
21	20	negeqd 11502	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋))
22	19, 21	ifbieq2d 4552	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0, +∞, -(log‘𝑥)) = if(𝑋 = 0, +∞, -(log‘𝑋)))
23		negex 11506	. . . . . . . . . . 11 ⊢ -(log‘𝑋) ∈ V
24	15, 23	ifex 4576	. . . . . . . . . 10 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V
25	22, 14, 24	fvmpt 7016	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)))
26		pnfxr 11315	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ∈ ℝ^*)
28		elunitrn 13507	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ)
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ)
30		elunitge0 33898	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (0[,]1) → 0 ≤ 𝑋)
31	30	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 ≤ 𝑋)
32		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → ¬ 𝑋 = 0)
33	32	neqned 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ≠ 0)
34	29, 31, 33	ne0gt0d 11398	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 < 𝑋)
35	29, 34	elrpd 13074	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ⁺)
36	35	relogcld 26665	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → (log‘𝑋) ∈ ℝ)
37	36	renegcld 11690	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ)
38	37	rexrd 11311	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ^*)
39	27, 38	ifclda 4561	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ ℝ^*)
40	25, 39	eqeltrd 2841	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ∈ ℝ^*)
41	40	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ∈ ℝ^*)
42		neeq1 3003	. . . . . . . . . 10 ⊢ (+∞ = if(𝑋 = 0, +∞, -(log‘𝑋)) → (+∞ ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
43		neeq1 3003	. . . . . . . . . 10 ⊢ (-(log‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)) → (-(log‘𝑋) ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
44		pnfnemnf 11316	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ≠ -∞)
46	37	renemnfd 11313	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ≠ -∞)
47	42, 43, 45, 46	ifbothda 4564	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞)
48	25, 47	eqnetrd 3008	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ≠ -∞)
49	48	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ≠ -∞)
50		xaddpnf1 13268	. . . . . . 7 ⊢ (((𝐹‘𝑋) ∈ ℝ^* ∧ (𝐹‘𝑋) ≠ -∞) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
51	41, 49, 50	syl2anc 584	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
52	18, 51	eqtrid 2789	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = +∞)
53		unitsscn 13540	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℂ
54		simpl 482	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ (0[,]1))
55	53, 54	sselid 3981	. . . . . . . 8 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ ℂ)
56	55	mul01d 11460	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 0) = 0)
57	56	fveq2d 6910	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = (𝐹‘0))
58	57, 17	eqtrdi 2793	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = +∞)
59	52, 58	eqtr4d 2780	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = (𝐹‘(𝑋 · 0)))
60		simpr 484	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑌 = 0)
61	60	fveq2d 6910	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑌) = (𝐹‘0))
62	61	oveq2d 7447	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘0)))
63	60	oveq2d 7447	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 𝑌) = (𝑋 · 0))
64	63	fveq2d 6910	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(𝑋 · 0)))
65	59, 62, 64	3eqtr4rd 2788	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
66	5	eleq2i 2833	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ 𝑋 ∈ (0[,]1))
67		elun 4153	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
68	66, 67	bitr3i 277	. . . . 5 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
69		elsni 4643	. . . . . 6 ⊢ (𝑋 ∈ {0} → 𝑋 = 0)
70	69	orim1i 910	. . . . 5 ⊢ ((𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
71	68, 70	sylbi 217	. . . 4 ⊢ (𝑋 ∈ (0[,]1) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
72	17	oveq1i 7441	. . . . . . . 8 ⊢ ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (+∞ +_𝑒 (𝐹‘𝑌))
73		simpr 484	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
74	14	xrge0iifcv 33933	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) = -(log‘𝑌))
75		0le0 12367	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
76		1re 11261	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
77		ltpnf 13162	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → 1 < +∞)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ 1 < +∞
79		iocssioo 13479	. . . . . . . . . . . . . . . . 17 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
80	1, 26, 75, 78, 79	mp4an 693	. . . . . . . . . . . . . . . 16 ⊢ (0(,]1) ⊆ (0(,)+∞)
81		ioorp 13465	. . . . . . . . . . . . . . . 16 ⊢ (0(,)+∞) = ℝ⁺
82	80, 81	sseqtri 4032	. . . . . . . . . . . . . . 15 ⊢ (0(,]1) ⊆ ℝ⁺
83	82	sseli 3979	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0(,]1) → 𝑌 ∈ ℝ⁺)
84	83	relogcld 26665	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0(,]1) → (log‘𝑌) ∈ ℝ)
85	84	renegcld 11690	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → -(log‘𝑌) ∈ ℝ)
86	74, 85	eqeltrd 2841	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ)
87	86	rexrd 11311	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ^*)
88	73, 87	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ∈ ℝ^*)
89	86	renemnfd 11313	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ≠ -∞)
90	73, 89	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ≠ -∞)
91		xaddpnf2 13269	. . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ℝ^* ∧ (𝐹‘𝑌) ≠ -∞) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
92	88, 90, 91	syl2anc 584	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
93	72, 92	eqtrid 2789	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = +∞)
94		rpssre 13042	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
95	82, 94	sstri 3993	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ
96		ax-resscn 11212	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
97	95, 96	sstri 3993	. . . . . . . . . . 11 ⊢ (0(,]1) ⊆ ℂ
98	97, 73	sselid 3981	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℂ)
99	98	mul02d 11459	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (0 · 𝑌) = 0)
100	99	fveq2d 6910	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = (𝐹‘0))
101	100, 17	eqtrdi 2793	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = +∞)
102	93, 101	eqtr4d 2780	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (𝐹‘(0 · 𝑌)))
103		simpl 482	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑋 = 0)
104	103	fveq2d 6910	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑋) = (𝐹‘0))
105	104	oveq1d 7446	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘0) +_𝑒 (𝐹‘𝑌)))
106	103	fvoveq1d 7453	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(0 · 𝑌)))
107	102, 105, 106	3eqtr4rd 2788	. . . . 5 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
108		simpl 482	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0(,]1))
109	82, 108	sselid 3981	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ⁺)
110	109	relogcld 26665	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℝ)
111	110	renegcld 11690	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑋) ∈ ℝ)
112		simpr 484	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
113	82, 112	sselid 3981	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ⁺)
114	113	relogcld 26665	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℝ)
115	114	renegcld 11690	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑌) ∈ ℝ)
116		rexadd 13274	. . . . . . 7 ⊢ ((-(log‘𝑋) ∈ ℝ ∧ -(log‘𝑌) ∈ ℝ) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
117	111, 115, 116	syl2anc 584	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
118	14	xrge0iifcv 33933	. . . . . . 7 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋))
119	118, 74	oveqan12d 7450	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = (-(log‘𝑋) +_𝑒 -(log‘𝑌)))
120	109	rpred 13077	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ)
121	113	rpred 13077	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ)
122	120, 121	remulcld 11291	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ ℝ)
123	109	rpgt0d 13080	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑋)
124	113	rpgt0d 13080	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑌)
125	120, 121, 123, 124	mulgt0d 11416	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < (𝑋 · 𝑌))
126		iocssicc 13477	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ (0[,]1)
127	126, 108	sselid 3981	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0[,]1))
128	126, 112	sselid 3981	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0[,]1))
129		iimulcl 24966	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
130	127, 128, 129	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
131		elicc01 13506	. . . . . . . . . . 11 ⊢ ((𝑋 · 𝑌) ∈ (0[,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 ≤ (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
132	131	simp3bi 1148	. . . . . . . . . 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 13450	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1)))
135	1, 76, 134	mp2an 692	. . . . . . . . 9 ⊢ ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
136	122, 125, 133, 135	syl3anbrc 1344	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0(,]1))
137	14	xrge0iifcv 33933	. . . . . . . 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 26667	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘(𝑋 · 𝑌)) = ((log‘𝑋) + (log‘𝑌)))
140	139	negeqd 11502	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘(𝑋 · 𝑌)) = -((log‘𝑋) + (log‘𝑌)))
141	110	recnd 11289	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℂ)
142	114	recnd 11289	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℂ)
143	141, 142	negdid 11633	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -((log‘𝑋) + (log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
144	138, 140, 143	3eqtrd 2781	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
145	117, 119, 144	3eqtr4rd 2788	. . . . 5 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
146	107, 145	jaoian 959	. . . 4 ⊢ (((𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
147	71, 146	sylan 580	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
148	65, 147	jaodan 960	. 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1))) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
149	11, 148	sylan2 593	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∪ cun 3949 ⊆ wss 3951 ifcif 4525 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 -cneg 11493 ℝ⁺crp 13034 +_𝑒 cxad 13152 (,)cioo 13387 (,]cioc 13388 [,]cicc 13390 ↾_t crest 17465 ordTopcordt 17544 logclog 26596

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598

This theorem is referenced by: xrge0iifmhm 33938 xrge0pluscn 33939

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