xrge0iifhom - Mathbox for Thierry Arnoux

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Theorem xrge0iifhom 33934

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 11228	. . . . . 6 ⊢ 0 ∈ ℝ^*
2		1xr 11240	. . . . . 6 ⊢ 1 ∈ ℝ^*
3		0le1 11708	. . . . . 6 ⊢ 0 ≤ 1
4		snunioc 13448	. . . . . 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 2821	. . . 4 ⊢ (𝑌 ∈ ({0} ∪ (0(,]1)) ↔ 𝑌 ∈ (0[,]1))
7		elun 4119	. . . 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 4609	. . . 4 ⊢ (𝑌 ∈ {0} → 𝑌 = 0)
10	9	orim1i 909	. . 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 13437	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
13		iftrue 4497	. . . . . . . . 9 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
14		xrge0iifhmeo.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
15		pnfex 11234	. . . . . . . . 9 ⊢ +∞ ∈ V
16	13, 14, 15	fvmpt 6971	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
17	12, 16	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) = +∞
18	17	oveq2i 7401	. . . . . 6 ⊢ ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = ((𝐹‘𝑋) +_𝑒 +∞)
19		eqeq1 2734	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0))
20		fveq2 6861	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋))
21	20	negeqd 11422	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋))
22	19, 21	ifbieq2d 4518	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0, +∞, -(log‘𝑥)) = if(𝑋 = 0, +∞, -(log‘𝑋)))
23		negex 11426	. . . . . . . . . . 11 ⊢ -(log‘𝑋) ∈ V
24	15, 23	ifex 4542	. . . . . . . . . 10 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V
25	22, 14, 24	fvmpt 6971	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)))
26		pnfxr 11235	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ∈ ℝ^*)
28		elunitrn 13435	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ)
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ)
30		elunitge0 33896	. . . . . . . . . . . . . . . 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 2933	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ≠ 0)
34	29, 31, 33	ne0gt0d 11318	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 0 < 𝑋)
35	29, 34	elrpd 12999	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ⁺)
36	35	relogcld 26539	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → (log‘𝑋) ∈ ℝ)
37	36	renegcld 11612	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ)
38	37	rexrd 11231	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ∈ ℝ^*)
39	27, 38	ifclda 4527	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ ℝ^*)
40	25, 39	eqeltrd 2829	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ∈ ℝ^*)
41	40	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ∈ ℝ^*)
42		neeq1 2988	. . . . . . . . . 10 ⊢ (+∞ = if(𝑋 = 0, +∞, -(log‘𝑋)) → (+∞ ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
43		neeq1 2988	. . . . . . . . . 10 ⊢ (-(log‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋)) → (-(log‘𝑋) ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞))
44		pnfnemnf 11236	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ≠ -∞)
46	37	renemnfd 11233	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 = 0) → -(log‘𝑋) ≠ -∞)
47	42, 43, 45, 46	ifbothda 4530	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) ≠ -∞)
48	25, 47	eqnetrd 2993	. . . . . . . 8 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ≠ -∞)
49	48	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ≠ -∞)
50		xaddpnf1 13193	. . . . . . 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 2777	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘0)) = +∞)
53		unitsscn 13468	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℂ
54		simpl 482	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ (0[,]1))
55	53, 54	sselid 3947	. . . . . . . 8 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ ℂ)
56	55	mul01d 11380	. . . . . . 7 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 0) = 0)
57	56	fveq2d 6865	. . . . . 6 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = (𝐹‘0))
58	57, 17	eqtrdi 2781	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = +∞)
59	52, 58	eqtr4d 2768	. . . 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 6865	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑌) = (𝐹‘0))
62	61	oveq2d 7406	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘0)))
63	60	oveq2d 7406	. . . . 5 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 𝑌) = (𝑋 · 0))
64	63	fveq2d 6865	. . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(𝑋 · 0)))
65	59, 62, 64	3eqtr4rd 2776	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
66	5	eleq2i 2821	. . . . . 6 ⊢ (𝑋 ∈ ({0} ∪ (0(,]1)) ↔ 𝑋 ∈ (0[,]1))
67		elun 4119	. . . . . 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 4609	. . . . . 6 ⊢ (𝑋 ∈ {0} → 𝑋 = 0)
70	69	orim1i 909	. . . . 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 7400	. . . . . . . 8 ⊢ ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = (+∞ +_𝑒 (𝐹‘𝑌))
73		simpr 484	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1))
74	14	xrge0iifcv 33931	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) = -(log‘𝑌))
75		0le0 12294	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
76		1re 11181	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
77		ltpnf 13087	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → 1 < +∞)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ 1 < +∞
79		iocssioo 13407	. . . . . . . . . . . . . . . . 17 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
80	1, 26, 75, 78, 79	mp4an 693	. . . . . . . . . . . . . . . 16 ⊢ (0(,]1) ⊆ (0(,)+∞)
81		ioorp 13393	. . . . . . . . . . . . . . . 16 ⊢ (0(,)+∞) = ℝ⁺
82	80, 81	sseqtri 3998	. . . . . . . . . . . . . . 15 ⊢ (0(,]1) ⊆ ℝ⁺
83	82	sseli 3945	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0(,]1) → 𝑌 ∈ ℝ⁺)
84	83	relogcld 26539	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0(,]1) → (log‘𝑌) ∈ ℝ)
85	84	renegcld 11612	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0(,]1) → -(log‘𝑌) ∈ ℝ)
86	74, 85	eqeltrd 2829	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ)
87	86	rexrd 11231	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ^*)
88	73, 87	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ∈ ℝ^*)
89	86	renemnfd 11233	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ≠ -∞)
90	73, 89	syl 17	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ≠ -∞)
91		xaddpnf2 13194	. . . . . . . . 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 2777	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +_𝑒 (𝐹‘𝑌)) = +∞)
94		rpssre 12966	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
95	82, 94	sstri 3959	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ
96		ax-resscn 11132	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
97	95, 96	sstri 3959	. . . . . . . . . . 11 ⊢ (0(,]1) ⊆ ℂ
98	97, 73	sselid 3947	. . . . . . . . . 10 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℂ)
99	98	mul02d 11379	. . . . . . . . 9 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (0 · 𝑌) = 0)
100	99	fveq2d 6865	. . . . . . . 8 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = (𝐹‘0))
101	100, 17	eqtrdi 2781	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = +∞)
102	93, 101	eqtr4d 2768	. . . . . 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 6865	. . . . . . 7 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑋) = (𝐹‘0))
105	104	oveq1d 7405	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = ((𝐹‘0) +_𝑒 (𝐹‘𝑌)))
106	103	fvoveq1d 7412	. . . . . 6 ⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(0 · 𝑌)))
107	102, 105, 106	3eqtr4rd 2776	. . . . 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 3947	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ⁺)
110	109	relogcld 26539	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℝ)
111	110	renegcld 11612	. . . . . . 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 3947	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ⁺)
114	113	relogcld 26539	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℝ)
115	114	renegcld 11612	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘𝑌) ∈ ℝ)
116		rexadd 13199	. . . . . . 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 33931	. . . . . . 7 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋))
119	118, 74	oveqan12d 7409	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)) = (-(log‘𝑋) +_𝑒 -(log‘𝑌)))
120	109	rpred 13002	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ ℝ)
121	113	rpred 13002	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℝ)
122	120, 121	remulcld 11211	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ ℝ)
123	109	rpgt0d 13005	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑋)
124	113	rpgt0d 13005	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < 𝑌)
125	120, 121, 123, 124	mulgt0d 11336	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 < (𝑋 · 𝑌))
126		iocssicc 13405	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ (0[,]1)
127	126, 108	sselid 3947	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈ (0[,]1))
128	126, 112	sselid 3947	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0[,]1))
129		iimulcl 24840	. . . . . . . . . . 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 13434	. . . . . . . . . . 11 ⊢ ((𝑋 · 𝑌) ∈ (0[,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 ≤ (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))
132	131	simp3bi 1147	. . . . . . . . . 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 13377	. . . . . . . . . 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 33931	. . . . . . . 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 26541	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘(𝑋 · 𝑌)) = ((log‘𝑋) + (log‘𝑌)))
140	139	negeqd 11422	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -(log‘(𝑋 · 𝑌)) = -((log‘𝑋) + (log‘𝑌)))
141	110	recnd 11209	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑋) ∈ ℂ)
142	114	recnd 11209	. . . . . . . 8 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (log‘𝑌) ∈ ℂ)
143	141, 142	negdid 11553	. . . . . . 7 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → -((log‘𝑋) + (log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
144	138, 140, 143	3eqtrd 2769	. . . . . 6 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (-(log‘𝑋) + -(log‘𝑌)))
145	117, 119, 144	3eqtr4rd 2776	. . . . 5 ⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +_𝑒 (𝐹‘𝑌)))
146	107, 145	jaoian 958	. . . 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 959	. 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 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∪ cun 3915 ⊆ wss 3917 ifcif 4491 {csn 4592 class class class wbr 5110 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 -cneg 11413 ℝ⁺crp 12958 +_𝑒 cxad 13077 (,)cioo 13313 (,]cioc 13314 [,]cicc 13316 ↾_t crest 17390 ordTopcordt 17469 logclog 26470

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-log 26472

This theorem is referenced by: xrge0iifmhm 33936 xrge0pluscn 33937

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