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Theorem cxple2 26661

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 1226	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
2		simpr 484	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 0 < 𝐴)
3	1, 2	elrpd 12983	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ⁺)
4	3	adantr 480	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐴 ∈ ℝ⁺)
5		simp2l 1201	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
6	5	ad2antrr 727	. . . . 5 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐵 ∈ ℝ)
7		simpr 484	. . . . 5 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 0 < 𝐵)
8	6, 7	elrpd 12983	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐵 ∈ ℝ⁺)
9		simp3 1139	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
10	9	ad2antrr 727	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐶 ∈ ℝ⁺)
11		simp3 1139	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
12	11	rpred 12986	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
13		relogcl 26539	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
14	13	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐴) ∈ ℝ)
15	12, 14	remulcld 11175	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐴)) ∈ ℝ)
16		relogcl 26539	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
17	16	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐵) ∈ ℝ)
18	12, 17	remulcld 11175	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐵)) ∈ ℝ)
19		efle 16085	. . . . . 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 585	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵)) ↔ (exp‘(𝐶 · (log‘𝐴))) ≤ (exp‘(𝐶 · (log‘𝐵)))))
21		efle 16085	. . . . . . 7 ⊢ (((log‘𝐴) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵))))
22	14, 17, 21	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵))))
23	14, 17, 11	lemul2d 13030	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵))))
24		reeflog 26544	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(log‘𝐴)) = 𝐴)
25	24	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐴)) = 𝐴)
26		reeflog 26544	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (exp‘(log‘𝐵)) = 𝐵)
27	26	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐵)) = 𝐵)
28	25, 27	breq12d 5098	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵)) ↔ 𝐴 ≤ 𝐵))
29	22, 23, 28	3bitr3rd 310	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ (𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵))))
30		rpre 12951	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
31	30	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
32	31	recnd 11173	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
33		rpne0 12959	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
34	33	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≠ 0)
35	12	recnd 11173	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
36		cxpef 26629	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
37	32, 34, 35, 36	syl3anc 1374	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
38		rpre 12951	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
39	38	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
40	39	recnd 11173	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
41		rpne0 12959	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ≠ 0)
42	41	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ≠ 0)
43		cxpef 26629	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
44	40, 42, 35, 43	syl3anc 1374	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
45	37, 44	breq12d 5098	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶) ↔ (exp‘(𝐶 · (log‘𝐴))) ≤ (exp‘(𝐶 · (log‘𝐵)))))
46	20, 29, 45	3bitr4d 311	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
47	4, 8, 10, 46	syl3anc 1374	. . 3 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
48		0re 11146	. . . . . . . 8 ⊢ 0 ∈ ℝ
49		simp1l 1199	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
50		ltnle 11225	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ ¬ 𝐴 ≤ 0))
51	48, 49, 50	sylancr 588	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 < 𝐴 ↔ ¬ 𝐴 ≤ 0))
52	51	biimpa 476	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ¬ 𝐴 ≤ 0)
53	9	rpred 12986	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
54	53	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → 𝐶 ∈ ℝ)
55		rpcxpcl 26640	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
56	3, 54, 55	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
57		rpgt0 12955	. . . . . . . . 9 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → 0 < (𝐴↑_𝑐𝐶))
58		rpre 12951	. . . . . . . . . 10 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → (𝐴↑_𝑐𝐶) ∈ ℝ)
59		ltnle 11225	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐴↑_𝑐𝐶) ∈ ℝ) → (0 < (𝐴↑_𝑐𝐶) ↔ ¬ (𝐴↑_𝑐𝐶) ≤ 0))
60	48, 58, 59	sylancr 588	. . . . . . . . 9 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → (0 < (𝐴↑_𝑐𝐶) ↔ ¬ (𝐴↑_𝑐𝐶) ≤ 0))
61	57, 60	mpbid 232	. . . . . . . 8 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → ¬ (𝐴↑_𝑐𝐶) ≤ 0)
62	56, 61	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ¬ (𝐴↑_𝑐𝐶) ≤ 0)
63	53	recnd 11173	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
64	9	rpne0d 12991	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
65		0cxp 26630	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (0↑_𝑐𝐶) = 0)
66	63, 64, 65	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0↑_𝑐𝐶) = 0)
67	66	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (0↑_𝑐𝐶) = 0)
68	67	breq2d 5097	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ((𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶) ↔ (𝐴↑_𝑐𝐶) ≤ 0))
69	62, 68	mtbird 325	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → ¬ (𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶))
70	52, 69	2falsed 376	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (𝐴 ≤ 0 ↔ (𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶)))
71		breq2 5089	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ≤ 0 ↔ 𝐴 ≤ 𝐵))
72		oveq1 7374	. . . . . . 7 ⊢ (0 = 𝐵 → (0↑_𝑐𝐶) = (𝐵↑_𝑐𝐶))
73	72	breq2d 5097	. . . . . 6 ⊢ (0 = 𝐵 → ((𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶) ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
74	71, 73	bibi12d 345	. . . . 5 ⊢ (0 = 𝐵 → ((𝐴 ≤ 0 ↔ (𝐴↑_𝑐𝐶) ≤ (0↑_𝑐𝐶)) ↔ (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶))))
75	70, 74	syl5ibcom 245	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (0 = 𝐵 → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶))))
76	75	imp 406	. . 3 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 = 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
77		simp2r 1202	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 0 ≤ 𝐵)
78		leloe 11232	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
79	48, 5, 78	sylancr 588	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
80	77, 79	mpbid 232	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 < 𝐵 ∨ 0 = 𝐵))
81	80	adantr 480	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (0 < 𝐵 ∨ 0 = 𝐵))
82	47, 76, 81	mpjaodan 961	. 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
83		simpr 484	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 = 𝐴)
84		simpl2r 1229	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 ≤ 𝐵)
85	83, 84	eqbrtrrd 5109	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 𝐴 ≤ 𝐵)
86	66	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (0↑_𝑐𝐶) = 0)
87	83	oveq1d 7382	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (0↑_𝑐𝐶) = (𝐴↑_𝑐𝐶))
88	86, 87	eqtr3d 2773	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 = (𝐴↑_𝑐𝐶))
89		simpl2l 1228	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 𝐵 ∈ ℝ)
90	53	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 𝐶 ∈ ℝ)
91		cxpge0 26647	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐶 ∈ ℝ) → 0 ≤ (𝐵↑_𝑐𝐶))
92	89, 84, 90, 91	syl3anc 1374	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 ≤ (𝐵↑_𝑐𝐶))
93	88, 92	eqbrtrrd 5109	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶))
94	85, 93	2thd 265	. 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
95		simp1r 1200	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 0 ≤ 𝐴)
96		leloe 11232	. . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
97	48, 49, 96	sylancr 588	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
98	95, 97	mpbid 232	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (0 < 𝐴 ∨ 0 = 𝐴))
99	82, 94, 98	mpjaodan 961	1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 · cmul 11043 < clt 11179 ≤ cle 11180 ℝ⁺crp 12942 expce 16026 logclog 26518 ↑_𝑐ccxp 26519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-cxp 26521

This theorem is referenced by: cxplt2 26662 cxple2a 26663 cxple2d 26691

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