cxple2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cxple2		Structured version Visualization version GIF version

Theorem cxple2 26942

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

**Assertion**
Ref	Expression
cxple2	⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))

**Proof of Theorem cxple2**
Step	Hyp	Ref	Expression
1		simpl1l 1243	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
2		simpr 490	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 0 < 𝐴)
3	1, 2	elrpd 13087	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ⁺)
4	3	adantr 486	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐴 ∈ ℝ⁺)
5		simp2l 1218	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
6	5	ad2antrr 739	. . . . 5 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐵 ∈ ℝ)
7		simpr 490	. . . . 5 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 0 < 𝐵)
8	6, 7	elrpd 13087	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐵 ∈ ℝ⁺)
9		simp3 1156	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
10	9	ad2antrr 739	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐶 ∈ ℝ⁺)
11		simp3 1156	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
12	11	rpred 13090	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
13		relogcl 26820	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
14	13	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐴) ∈ ℝ)
15	12, 14	remulcld 11267	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐴)) ∈ ℝ)
16		relogcl 26820	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
17	16	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐵) ∈ ℝ)
18	12, 17	remulcld 11267	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐵)) ∈ ℝ)
19		efle 16212	. . . . . 6 ⊢ (((𝐶 · (log‘𝐴)) ∈ ℝ ∧ (𝐶 · (log‘𝐵)) ∈ ℝ) → ((𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵)) ↔ (exp‘(𝐶 · (log‘𝐴))) ≤ (exp‘(𝐶 · (log‘𝐵)))))
20	15, 18, 19	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵)) ↔ (exp‘(𝐶 · (log‘𝐴))) ≤ (exp‘(𝐶 · (log‘𝐵)))))
21		efle 16212	. . . . . . 7 ⊢ (((log‘𝐴) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵))))
22	14, 17, 21	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵))))
23	14, 17, 11	lemul2d 13134	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵))))
24		reeflog 26825	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(log‘𝐴)) = 𝐴)
25	24	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐴)) = 𝐴)
26		reeflog 26825	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (exp‘(log‘𝐵)) = 𝐵)
27	26	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐵)) = 𝐵)
28	25, 27	breq12d 5120	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵)) ↔ 𝐴 ≤ 𝐵))
29	22, 23, 28	3bitr3rd 313	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ (𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵))))
30		rpre 13055	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
31	30	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
32	31	recnd 11265	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
33		rpne0 13063	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
34	33	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≠ 0)
35	12	recnd 11265	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
36		cxpef 26910	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
37	32, 34, 35, 36	syl3anc 1398	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
38		rpre 13055	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
39	38	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
40	39	recnd 11265	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
41		rpne0 13063	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ≠ 0)
42	41	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ≠ 0)
43		cxpef 26910	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
44	40, 42, 35, 43	syl3anc 1398	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
45	37, 44	breq12d 5120	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶) ↔ (exp‘(𝐶 · (log‘𝐴))) ≤ (exp‘(𝐶 · (log‘𝐵)))))
46	20, 29, 45	3bitr4d 314	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
47	4, 8, 10, 46	syl3anc 1398	. . 3 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
48		0re 11238	. . . . . . . 8 ⊢ 0 ∈ ℝ
49		simp1l 1216	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
50		ltnle 11317	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ ¬ 𝐴 ≤ 0))
51	48, 49, 50	sylancr 599	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 < 𝐴 ↔ ¬ 𝐴 ≤ 0))
52	51	biimpa 482	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ¬ 𝐴 ≤ 0)
53	9	rpred 13090	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
54	53	adantr 486	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 𝐶 ∈ ℝ)
55		rpcxpcl 26921	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
56	3, 54, 55	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
57		rpgt0 13059	. . . . . . . . 9 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → 0 < (𝐴↑_𝑐𝐶))
58		rpre 13055	. . . . . . . . . 10 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → (𝐴↑_𝑐𝐶) ∈ ℝ)
59		ltnle 11317	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐴↑_𝑐𝐶) ∈ ℝ) → (0 < (𝐴↑_𝑐𝐶) ↔ ¬ (𝐴↑_𝑐𝐶) ≤ 0))
60	48, 58, 59	sylancr 599	. . . . . . . . 9 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → (0 < (𝐴↑_𝑐𝐶) ↔ ¬ (𝐴↑_𝑐𝐶) ≤ 0))
61	57, 60	mpbid 235	. . . . . . . 8 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → ¬ (𝐴↑_𝑐𝐶) ≤ 0)
62	56, 61	syl 18	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ¬ (𝐴↑_𝑐𝐶) ≤ 0)
63	53	recnd 11265	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
64	9	rpne0d 13095	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
65		0cxp 26911	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (0↑_𝑐𝐶) = 0)
66	63, 64, 65	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0↑_𝑐𝐶) = 0)
67	66	adantr 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (0↑_𝑐𝐶) = 0)
68	67	breq2d 5119	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ((𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶) ↔ (𝐴↑_𝑐𝐶) ≤ 0))
69	62, 68	mtbird 328	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ¬ (𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶))
70	52, 69	2falsed 379	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (𝐴 ≤ 0 ↔ (𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶)))
71		breq2 5111	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ≤ 0 ↔ 𝐴 ≤ 𝐵))
72		oveq1 7424	. . . . . . 7 ⊢ (0 = 𝐵 → (0↑_𝑐𝐶) = (𝐵↑_𝑐𝐶))
73	72	breq2d 5119	. . . . . 6 ⊢ (0 = 𝐵 → ((𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶) ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
74	71, 73	bibi12d 348	. . . . 5 ⊢ (0 = 𝐵 → ((𝐴 ≤ 0 ↔ (𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶)) ↔ (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶))))
75	70, 74	syl5ibcom 248	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (0 = 𝐵 → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶))))
76	75	imp 412	. . 3 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 = 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
77		simp2r 1219	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 0 ≤ 𝐵)
78		leloe 11324	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
79	48, 5, 78	sylancr 599	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
80	77, 79	mpbid 235	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 < 𝐵 ∨ 0 = 𝐵))
81	80	adantr 486	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (0 < 𝐵 ∨ 0 = 𝐵))
82	47, 76, 81	mpjaodan 973	. 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
83		simpr 490	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 = 𝐴)
84		simpl2r 1246	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 ≤ 𝐵)
85	83, 84	eqbrtrrd 5133	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 𝐴 ≤ 𝐵)
86	66	adantr 486	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (0↑_𝑐𝐶) = 0)
87	83	oveq1d 7432	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (0↑_𝑐𝐶) = (𝐴↑_𝑐𝐶))
88	86, 87	eqtr3d 2799	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 = (𝐴↑_𝑐𝐶))
89		simpl2l 1245	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 𝐵 ∈ ℝ)
90	53	adantr 486	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 𝐶 ∈ ℝ)
91		cxpge0 26928	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐶 ∈ ℝ) → 0 ≤ (𝐵↑_𝑐𝐶))
92	89, 84, 90, 91	syl3anc 1398	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 ≤ (𝐵↑_𝑐𝐶))
93	88, 92	eqbrtrrd 5133	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶))
94	85, 93	2thd 268	. 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
95		simp1r 1217	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 0 ≤ 𝐴)
96		leloe 11324	. . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
97	48, 49, 96	sylancr 599	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
98	95, 97	mpbid 235	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 < 𝐴 ∨ 0 = 𝐴))
99	82, 94, 98	mpjaodan 973	1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 ℝcr 11127 0cc0 11128 · cmul 11133 < clt 11271 ≤ cle 11272 ℝ⁺crp 13046 expce 16153 logclog 26799 ↑_𝑐ccxp 26800

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ioc 13407 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-fl 13857 df-mod 13935 df-seq 14070 df-exp 14130 df-fac 14342 df-bc 14371 df-hash 14399 df-shft 15144 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-limsup 15562 df-clim 15579 df-rlim 15580 df-sum 15778 df-ef 16159 df-sin 16161 df-cos 16162 df-pi 16164 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-rest 17513 df-topn 17514 df-0g 17532 df-gsum 17533 df-topgen 17534 df-pt 17535 df-prds 17538 df-xrs 17594 df-qtop 17599 df-imas 17600 df-xps 17602 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-mulg 19197 df-cntz 19450 df-cmn 19915 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-fbas 21588 df-fg 21589 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-nei 23329 df-lp 23367 df-perf 23368 df-cn 23458 df-cnp 23459 df-haus 23546 df-tx 23794 df-hmeo 23987 df-fil 24078 df-fm 24170 df-flim 24171 df-flf 24172 df-xms 24552 df-ms 24553 df-tms 24554 df-cncf 25112 df-limc 26100 df-dv 26101 df-log 26801 df-cxp 26802

This theorem is used by: cxplt2 26943 cxple2a 26944 cxple2d 26972

W3C validator