xrge0iifhom - Mathbox for Thierry Arnoux

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Theorem xrge0iifhom 34336

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 11262	. . . . . 6 ⊢ 0 ∈ ℝ^*
2		1xr 11274	. . . . . 6 ⊢ 1 ∈ ℝ^*
3		0le1 11743	. . . . . 6 ⊢ 0 ≤ 1
4		snunioc 13513	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
5	1, 2, 3, 4	mp3an 1489	. . . . 5 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
6	5	eleq2i 2854	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ 𝑌 ∈ (0[,]1))
7		elun 4106	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
8	6, 7	bitr3i 280	. . 3 ⊢ (𝑌 ∈ (0[,]1) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
9		elsni 4605	. . . 4 ⊢ (𝑌 ∈ {0} → 𝑌 = 0)
10	9	orim1i 922	. . 3 ⊢ ((𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
11	8, 10	sylbi 220	. 2 ⊢ (𝑌 ∈ (0[,]1) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
12		0elunit 13502	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
13		iftrue 4492	. . . . . . . . 9 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
14		xrge0iifhmeo.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
15		pnfex 11268	. . . . . . . . 9 ⊢ +∞ ∈ V
16	13, 14, 15	fvmpt 6989	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
17	12, 16	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) = +∞
18	17	oveq2i 7423	. . . . . 6 ⊢ ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = ((𝐹‘𝑋) +_𝑒 +∞)
19		eqeq1 2766	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0))
20		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋))
21	20	negeqd 11457	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋))
22	19, 21	ifbieq2d 4513	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0, +∞, -(log‘𝑥)) = if(𝑋 = 0, +∞, -(log‘𝑋)))
23		negex 11461	. . . . . . . . . . 11 ⊢ -(log‘𝑋) ∈ V
24	15, 23	ifex 4537	. . . . . . . . . 10 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V
25	22, 14, 24	fvmpt 6989	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)))
26		pnfxr 11269	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ∈ ℝ^*)
28		elunitrn 13500	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ)
29	28	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ)
30		elunitge0 34298	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (0[,]1) → 0 ≤ 𝑋)
31	30	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 ≤ 𝑋)
32		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → ¬ 𝑋 = 0)
33	32	neqned 2964	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ≠ 0)
34	29, 31, 33	ne0gt0d 11353	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 < 𝑋)
35	29, 34	elrpd 13063	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ⁺)
36	35	relogcld 26799	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → (log‘𝑋) ∈ ℝ)
37	36	renegcld 11647	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ)
38	37	rexrd 11265	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ^*)
39	27, 38	ifclda 4522	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ ℝ^*)
40	25, 39	eqeltrd 2862	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ∈ ℝ^*)
41	40	adantr 485	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ∈ ℝ^*)
42		neeq1 3019	. . . . . . . . . 10 ⊢ (+∞ = if(𝑋 = 0, +∞, -(log‘𝑋)) → (+∞ ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
43		neeq1 3019	. . . . . . . . . 10 ⊢ (-(log‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)) → (-(log‘𝑋) ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
44		pnfnemnf 11270	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ≠ -∞)
46	37	renemnfd 11267	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ≠ -∞)
47	42, 43, 45, 46	ifbothda 4525	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞)
48	25, 47	eqnetrd 3024	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ≠ -∞)
49	48	adantr 485	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ≠ -∞)
50		xaddpnf1 13258	. . . . . . 7 ⊢ (((𝐹‘𝑋) ∈ ℝ^* ∧ (𝐹‘𝑋) ≠ -∞) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
51	41, 49, 50	syl2anc 595	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
52	18, 51	eqtrid 2809	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = +∞)
53		unitsscn 13533	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℂ
54		simpl 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ (0[,]1))
55	53, 54	sselid 3934	. . . . . . . 8 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ ℂ)
56	55	mul01d 11415	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 0) = 0)
57	56	fveq2d 6885	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = (𝐹‘0))
58	57, 17	eqtrdi 2813	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = +∞)
59	52, 58	eqtr4d 2800	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = (𝐹‘(𝑋 · 0)))
60		simpr 489	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑌 = 0)
61	60	fveq2d 6885	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑌) = (𝐹‘0))
62	61	oveq2d 7428	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘0)))
63	60	oveq2d 7428	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 𝑌) = (𝑋 · 0))
64	63	fveq2d 6885	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(𝑋 · 0)))
65	59, 62, 64	3eqtr4rd 2808	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
66	5	eleq2i 2854	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ 𝑋 ∈ (0[,]1))
67		elun 4106	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
68	66, 67	bitr3i 280	. . . . 5 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
69		elsni 4605	. . . . . 6 ⊢ (𝑋 ∈ {0} → 𝑋 = 0)
70	69	orim1i 922	. . . . 5 ⊢ ((𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
71	68, 70	sylbi 220	. . . 4 ⊢ (𝑋 ∈ (0[,]1) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
72	17	oveq1i 7422	. . . . . . . 8 ⊢ ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (+∞ +_𝑒 (𝐹‘𝑌))
73		simpr 489	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
74	14	xrge0iifcv 34333	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) = -(log‘𝑌))
75		0le0 12348	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
76		1re 11214	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
77		ltpnf 13151	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → 1 < +∞)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ 1 < +∞
79		iocssioo 13472	. . . . . . . . . . . . . . . . 17 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
80	1, 26, 75, 78, 79	mp4an 705	. . . . . . . . . . . . . . . 16 ⊢ (0(,]1) ⊆ (0(,)+∞)
81		ioorp 13458	. . . . . . . . . . . . . . . 16 ⊢ (0(,)+∞) = ℝ⁺
82	80, 81	sseqtri 3984	. . . . . . . . . . . . . . 15 ⊢ (0(,]1) ⊆ ℝ⁺
83	82	sseli 3932	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0(,]1) → 𝑌 ∈ ℝ⁺)
84	83	relogcld 26799	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0(,]1) → (log‘𝑌) ∈ ℝ)
85	84	renegcld 11647	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → -(log‘𝑌) ∈ ℝ)
86	74, 85	eqeltrd 2862	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ)
87	86	rexrd 11265	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ^*)
88	73, 87	syl 18	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ∈ ℝ^*)
89	86	renemnfd 11267	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ≠ -∞)
90	73, 89	syl 18	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ≠ -∞)
91		xaddpnf2 13259	. . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ℝ^* ∧ (𝐹‘𝑌) ≠ -∞) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
92	88, 90, 91	syl2anc 595	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
93	72, 92	eqtrid 2809	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = +∞)
94		rpssre 13030	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
95	82, 94	sstri 3945	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ
96		ax-resscn 11163	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
97	95, 96	sstri 3945	. . . . . . . . . . 11 ⊢ (0(,]1) ⊆ ℂ
98	97, 73	sselid 3934	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℂ)
99	98	mul02d 11414	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (0 · 𝑌) = 0)
100	99	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = (𝐹‘0))
101	100, 17	eqtrdi 2813	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = +∞)
102	93, 101	eqtr4d 2800	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (𝐹‘(0 · 𝑌)))
103		simpl 487	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑋 = 0)
104	103	fveq2d 6885	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑋) = (𝐹‘0))
105	104	oveq1d 7427	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘0) +_𝑒 (𝐹‘𝑌)))
106	103	fvoveq1d 7434	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(0 · 𝑌)))
107	102, 105, 106	3eqtr4rd 2808	. . . . 5 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
108		simpl 487	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0(,]1))
109	82, 108	sselid 3934	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ⁺)
110	109	relogcld 26799	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℝ)
111	110	renegcld 11647	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑋) ∈ ℝ)
112		simpr 489	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
113	82, 112	sselid 3934	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ⁺)
114	113	relogcld 26799	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℝ)
115	114	renegcld 11647	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑌) ∈ ℝ)
116		rexadd 13264	. . . . . . 7 ⊢ ((-(log‘𝑋) ∈ ℝ ∧ -(log‘𝑌) ∈ ℝ) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
117	111, 115, 116	syl2anc 595	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
118	14	xrge0iifcv 34333	. . . . . . 7 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋))
119	118, 74	oveqan12d 7431	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = (-(log‘𝑋) +_𝑒 -(log‘𝑌)))
120	109	rpred 13066	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ)
121	113	rpred 13066	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ)
122	120, 121	remulcld 11245	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ ℝ)
123	109	rpgt0d 13069	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑋)
124	113	rpgt0d 13069	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑌)
125	120, 121, 123, 124	mulgt0d 11371	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < (𝑋 · 𝑌))
126		iocssicc 13470	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ (0[,]1)
127	126, 108	sselid 3934	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0[,]1))
128	126, 112	sselid 3934	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0[,]1))
129		iimulcl 25107	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
130	127, 128, 129	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
131		elicc01 13499	. . . . . . . . . . 11 ⊢ ((𝑋 · 𝑌) ∈ (0[,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 ≤ (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
132	131	simp3bi 1164	. . . . . . . . . 10 ⊢ ((𝑋 · 𝑌) ∈ (0[,]1) → (𝑋 · 𝑌) ≤ 1)
133	130, 132	syl 18	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ≤ 1)
134		elioc2 13442	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1)))
135	1, 76, 134	mp2an 704	. . . . . . . . 9 ⊢ ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
136	122, 125, 133, 135	syl3anbrc 1361	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0(,]1))
137	14	xrge0iifcv 34333	. . . . . . . 8 ⊢ ((𝑋 · 𝑌) ∈ (0(,]1) → (𝐹‘(𝑋 · 𝑌)) = -(log‘(𝑋 · 𝑌)))
138	136, 137	syl 18	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = -(log‘(𝑋 · 𝑌)))
139	109, 113	relogmuld 26801	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘(𝑋 · 𝑌)) = ((log‘𝑋) + (log‘𝑌)))
140	139	negeqd 11457	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘(𝑋 · 𝑌)) = -((log‘𝑋) + (log‘𝑌)))
141	110	recnd 11243	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℂ)
142	114	recnd 11243	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℂ)
143	141, 142	negdid 11588	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -((log‘𝑋) + (log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
144	138, 140, 143	3eqtrd 2801	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
145	117, 119, 144	3eqtr4rd 2808	. . . . 5 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
146	107, 145	jaoian 970	. . . 4 ⊢ (((𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
147	71, 146	sylan 591	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
148	65, 147	jaodan 971	. 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1))) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
149	11, 148	sylan2 604	1 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∪ cun 3902 ⊆ wss 3904 ifcif 4486 {csn 4588 class class class wbr 5108 ↦ cmpt 5191 ‘cfv 6536 (class class class)co 7412 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 +∞cpnf 11246 -∞cmnf 11247 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 -cneg 11448 ℝ⁺crp 13022 +_𝑒 cxad 13141 (,)cioo 13378 (,]cioc 13379 [,]cicc 13381 ↾_t crest 17479 ordTopcordt 17559 logclog 26730

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732

This theorem is used by: xrge0iifmhm 34338 xrge0pluscn 34339

W3C validator