xrge0iifhom - Mathbox for Thierry Arnoux

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Theorem xrge0iifhom 30857

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 10486	. . . . . 6 ⊢ 0 ∈ ℝ^*
2		1xr 10499	. . . . . 6 ⊢ 1 ∈ ℝ^*
3		0le1 10963	. . . . . 6 ⊢ 0 ≤ 1
4		snunioc 12681	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
5	1, 2, 3, 4	mp3an 1441	. . . . 5 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
6	5	eleq2i 2852	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ 𝑌 ∈ (0[,]1))
7		elun 4009	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
8	6, 7	bitr3i 269	. . 3 ⊢ (𝑌 ∈ (0[,]1) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)))
9		elsni 4453	. . . 4 ⊢ (𝑌 ∈ {0} → 𝑌 = 0)
10	9	orim1i 894	. . 3 ⊢ ((𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
11	8, 10	sylbi 209	. 2 ⊢ (𝑌 ∈ (0[,]1) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1)))
12		0elunit 12670	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
13		iftrue 4351	. . . . . . . . 9 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
14		xrge0iifhmeo.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
15		pnfex 10492	. . . . . . . . 9 ⊢ +∞ ∈ V
16	13, 14, 15	fvmpt 6594	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
17	12, 16	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) = +∞
18	17	oveq2i 6986	. . . . . 6 ⊢ ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = ((𝐹‘𝑋) +_𝑒 +∞)
19		eqeq1 2777	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0))
20		fveq2 6497	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋))
21	20	negeqd 10679	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋))
22	19, 21	ifbieq2d 4370	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0, +∞, -(log‘𝑥)) = if(𝑋 = 0, +∞, -(log‘𝑋)))
23		negex 10683	. . . . . . . . . . 11 ⊢ -(log‘𝑋) ∈ V
24	15, 23	ifex 4393	. . . . . . . . . 10 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V
25	22, 14, 24	fvmpt 6594	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)))
26		pnfxr 10493	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ∈ ℝ^*)
28		elunitrn 30817	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ)
29	28	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ)
30		elunitge0 30819	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (0[,]1) → 0 ≤ 𝑋)
31	30	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 ≤ 𝑋)
32		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → ¬ 𝑋 = 0)
33	32	neqned 2969	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ≠ 0)
34	29, 31, 33	ne0gt0d 10576	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 < 𝑋)
35	29, 34	elrpd 12244	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ⁺)
36	35	relogcld 24923	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → (log‘𝑋) ∈ ℝ)
37	36	renegcld 10867	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ)
38	37	rexrd 10489	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ^*)
39	27, 38	ifclda 4379	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ ℝ^*)
40	25, 39	eqeltrd 2861	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ∈ ℝ^*)
41	40	adantr 473	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ∈ ℝ^*)
42		neeq1 3024	. . . . . . . . . 10 ⊢ (+∞ = if(𝑋 = 0, +∞, -(log‘𝑋)) → (+∞ ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
43		neeq1 3024	. . . . . . . . . 10 ⊢ (-(log‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)) → (-(log‘𝑋) ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
44		pnfnemnf 10495	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ≠ -∞)
46	37	renemnfd 10491	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ≠ -∞)
47	42, 43, 45, 46	ifbothda 4382	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞)
48	25, 47	eqnetrd 3029	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ≠ -∞)
49	48	adantr 473	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ≠ -∞)
50		xaddpnf1 12435	. . . . . . 7 ⊢ (((𝐹‘𝑋) ∈ ℝ^* ∧ (𝐹‘𝑋) ≠ -∞) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
51	41, 49, 50	syl2anc 576	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 +∞) = +∞)
52	18, 51	syl5eq 2821	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = +∞)
53		unitsscn 30816	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℂ
54		simpl 475	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ (0[,]1))
55	53, 54	sseldi 3851	. . . . . . . 8 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ ℂ)
56	55	mul01d 10638	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 0) = 0)
57	56	fveq2d 6501	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = (𝐹‘0))
58	57, 17	syl6eq 2825	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = +∞)
59	52, 58	eqtr4d 2812	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = (𝐹‘(𝑋 · 0)))
60		simpr 477	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑌 = 0)
61	60	fveq2d 6501	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑌) = (𝐹‘0))
62	61	oveq2d 6991	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘0)))
63	60	oveq2d 6991	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 𝑌) = (𝑋 · 0))
64	63	fveq2d 6501	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(𝑋 · 0)))
65	59, 62, 64	3eqtr4rd 2820	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
66	5	eleq2i 2852	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ 𝑋 ∈ (0[,]1))
67		elun 4009	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
68	66, 67	bitr3i 269	. . . . 5 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)))
69		elsni 4453	. . . . . 6 ⊢ (𝑋 ∈ {0} → 𝑋 = 0)
70	69	orim1i 894	. . . . 5 ⊢ ((𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
71	68, 70	sylbi 209	. . . 4 ⊢ (𝑋 ∈ (0[,]1) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)))
72	17	oveq1i 6985	. . . . . . . 8 ⊢ ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (+∞ +_𝑒 (𝐹‘𝑌))
73		simpr 477	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
74	14	xrge0iifcv 30854	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) = -(log‘𝑌))
75		0le0 11547	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
76		1re 10438	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
77		ltpnf 12331	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → 1 < +∞)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ 1 < +∞
79		iocssioo 12642	. . . . . . . . . . . . . . . . 17 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
80	1, 26, 75, 78, 79	mp4an 681	. . . . . . . . . . . . . . . 16 ⊢ (0(,]1) ⊆ (0(,)+∞)
81		ioorp 12629	. . . . . . . . . . . . . . . 16 ⊢ (0(,)+∞) = ℝ⁺
82	80, 81	sseqtri 3888	. . . . . . . . . . . . . . 15 ⊢ (0(,]1) ⊆ ℝ⁺
83	82	sseli 3849	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0(,]1) → 𝑌 ∈ ℝ⁺)
84	83	relogcld 24923	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0(,]1) → (log‘𝑌) ∈ ℝ)
85	84	renegcld 10867	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → -(log‘𝑌) ∈ ℝ)
86	74, 85	eqeltrd 2861	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ)
87	86	rexrd 10489	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ^*)
88	73, 87	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ∈ ℝ^*)
89	86	renemnfd 10491	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ≠ -∞)
90	73, 89	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ≠ -∞)
91		xaddpnf2 12436	. . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ℝ^* ∧ (𝐹‘𝑌) ≠ -∞) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
92	88, 90, 91	syl2anc 576	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (+∞ +_𝑒 (𝐹‘𝑌)) = +∞)
93	72, 92	syl5eq 2821	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = +∞)
94		rpssre 12210	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
95	82, 94	sstri 3862	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ
96		ax-resscn 10391	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
97	95, 96	sstri 3862	. . . . . . . . . . 11 ⊢ (0(,]1) ⊆ ℂ
98	97, 73	sseldi 3851	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℂ)
99	98	mul02d 10637	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (0 · 𝑌) = 0)
100	99	fveq2d 6501	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = (𝐹‘0))
101	100, 17	syl6eq 2825	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = +∞)
102	93, 101	eqtr4d 2812	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (𝐹‘(0 · 𝑌)))
103		simpl 475	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑋 = 0)
104	103	fveq2d 6501	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑋) = (𝐹‘0))
105	104	oveq1d 6990	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘0) +_𝑒 (𝐹‘𝑌)))
106	103	fvoveq1d 6997	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(0 · 𝑌)))
107	102, 105, 106	3eqtr4rd 2820	. . . . 5 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
108		simpl 475	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0(,]1))
109	82, 108	sseldi 3851	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ⁺)
110	109	relogcld 24923	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℝ)
111	110	renegcld 10867	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑋) ∈ ℝ)
112		simpr 477	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
113	82, 112	sseldi 3851	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ⁺)
114	113	relogcld 24923	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℝ)
115	114	renegcld 10867	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑌) ∈ ℝ)
116		rexadd 12441	. . . . . . 7 ⊢ ((-(log‘𝑋) ∈ ℝ ∧ -(log‘𝑌) ∈ ℝ) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
117	111, 115, 116	syl2anc 576	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (-(log‘𝑋) +_𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
118	14	xrge0iifcv 30854	. . . . . . 7 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋))
119	118, 74	oveqan12d 6994	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = (-(log‘𝑋) +_𝑒 -(log‘𝑌)))
120	109	rpred 12247	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ)
121	113	rpred 12247	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ)
122	120, 121	remulcld 10469	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ ℝ)
123	109	rpgt0d 12250	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑋)
124	113	rpgt0d 12250	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑌)
125	120, 121, 123, 124	mulgt0d 10594	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < (𝑋 · 𝑌))
126		iocssicc 12640	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ (0[,]1)
127	126, 108	sseldi 3851	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0[,]1))
128	126, 112	sseldi 3851	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0[,]1))
129		iimulcl 23260	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
130	127, 128, 129	syl2anc 576	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0[,]1))
131		elicc01 12669	. . . . . . . . . . 11 ⊢ ((𝑋 · 𝑌) ∈ (0[,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 ≤ (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
132	131	simp3bi 1128	. . . . . . . . . 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 12614	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1)))
135	1, 76, 134	mp2an 680	. . . . . . . . 9 ⊢ ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
136	122, 125, 133, 135	syl3anbrc 1324	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0(,]1))
137	14	xrge0iifcv 30854	. . . . . . . 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 24925	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘(𝑋 · 𝑌)) = ((log‘𝑋) + (log‘𝑌)))
140	139	negeqd 10679	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘(𝑋 · 𝑌)) = -((log‘𝑋) + (log‘𝑌)))
141	110	recnd 10467	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℂ)
142	114	recnd 10467	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℂ)
143	141, 142	negdid 10810	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -((log‘𝑋) + (log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
144	138, 140, 143	3eqtrd 2813	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
145	117, 119, 144	3eqtr4rd 2820	. . . . 5 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
146	107, 145	jaoian 940	. . . 4 ⊢ (((𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
147	71, 146	sylan 572	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
148	65, 147	jaodan 941	. 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1))) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
149	11, 148	sylan2 584	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 198 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∪ cun 3822 ⊆ wss 3824 ifcif 4345 {csn 4436 class class class wbr 4926 ↦ cmpt 5005 ‘cfv 6186 (class class class)co 6975 ℂcc 10332 ℝcr 10333 0cc0 10334 1c1 10335 + caddc 10337 · cmul 10339 +∞cpnf 10470 -∞cmnf 10471 ℝ^*cxr 10472 < clt 10473 ≤ cle 10474 -cneg 10670 ℝ⁺crp 12203 +_𝑒 cxad 12321 (,)cioo 12553 (,]cioc 12554 [,]cicc 12556 ↾_t crest 16549 ordTopcordt 16627 logclog 24855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 ax-addf 10413 ax-mulf 10414

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-iin 4792 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-of 7226 df-om 7396 df-1st 7500 df-2nd 7501 df-supp 7633 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-ixp 8259 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-fsupp 8628 df-fi 8669 df-sup 8700 df-inf 8701 df-oi 8768 df-card 9161 df-cda 9387 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-z 11793 df-dec 11911 df-uz 12058 df-q 12162 df-rp 12204 df-xneg 12323 df-xadd 12324 df-xmul 12325 df-ioo 12557 df-ioc 12558 df-ico 12559 df-icc 12560 df-fz 12708 df-fzo 12849 df-fl 12976 df-mod 13052 df-seq 13184 df-exp 13244 df-fac 13448 df-bc 13477 df-hash 13505 df-shft 14286 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-limsup 14688 df-clim 14705 df-rlim 14706 df-sum 14903 df-ef 15280 df-sin 15282 df-cos 15283 df-pi 15285 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-starv 16435 df-sca 16436 df-vsca 16437 df-ip 16438 df-tset 16439 df-ple 16440 df-ds 16442 df-unif 16443 df-hom 16444 df-cco 16445 df-rest 16551 df-topn 16552 df-0g 16570 df-gsum 16571 df-topgen 16572 df-pt 16573 df-prds 16576 df-xrs 16630 df-qtop 16635 df-imas 16636 df-xps 16638 df-mre 16728 df-mrc 16729 df-acs 16731 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-submnd 17817 df-mulg 18025 df-cntz 18231 df-cmn 18681 df-psmet 20255 df-xmet 20256 df-met 20257 df-bl 20258 df-mopn 20259 df-fbas 20260 df-fg 20261 df-cnfld 20264 df-top 21222 df-topon 21239 df-topsp 21261 df-bases 21274 df-cld 21347 df-ntr 21348 df-cls 21349 df-nei 21426 df-lp 21464 df-perf 21465 df-cn 21555 df-cnp 21556 df-haus 21643 df-tx 21890 df-hmeo 22083 df-fil 22174 df-fm 22266 df-flim 22267 df-flf 22268 df-xms 22649 df-ms 22650 df-tms 22651 df-cncf 23205 df-limc 24183 df-dv 24184 df-log 24857

This theorem is referenced by: xrge0iifmhm 30859 xrge0pluscn 30860

W3C validator