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Theorem cxple2 26658

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 1225	. . . . . 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 13048	. . . . 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 1200	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
6	5	ad2antrr 726	. . . . 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 13048	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐵 ∈ ℝ⁺)
9		simp3 1138	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
10	9	ad2antrr 726	. . . 4 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → 𝐶 ∈ ℝ⁺)
11		simp3 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
12	11	rpred 13051	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
13		relogcl 26536	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
14	13	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐴) ∈ ℝ)
15	12, 14	remulcld 11265	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐴)) ∈ ℝ)
16		relogcl 26536	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
17	16	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐵) ∈ ℝ)
18	12, 17	remulcld 11265	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐵)) ∈ ℝ)
19		efle 16136	. . . . . 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 584	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵)) ↔ (exp‘(𝐶 · (log‘𝐴))) ≤ (exp‘(𝐶 · (log‘𝐵)))))
21		efle 16136	. . . . . . 7 ⊢ (((log‘𝐴) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵))))
22	14, 17, 21	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (exp‘(log‘𝐴)) ≤ (exp‘(log‘𝐵))))
23	14, 17, 11	lemul2d 13095	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵))))
24		reeflog 26541	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(log‘𝐴)) = 𝐴)
25	24	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐴)) = 𝐴)
26		reeflog 26541	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (exp‘(log‘𝐵)) = 𝐵)
27	26	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐵)) = 𝐵)
28	25, 27	breq12d 5132	. . . . . 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 13017	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
31	30	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
32	31	recnd 11263	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
33		rpne0 13025	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
34	33	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≠ 0)
35	12	recnd 11263	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
36		cxpef 26626	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
37	32, 34, 35, 36	syl3anc 1373	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
38		rpre 13017	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
39	38	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
40	39	recnd 11263	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
41		rpne0 13025	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ≠ 0)
42	41	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ≠ 0)
43		cxpef 26626	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
44	40, 42, 35, 43	syl3anc 1373	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
45	37, 44	breq12d 5132	. . . . 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 1373	. . 3 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
48		0re 11237	. . . . . . . 8 ⊢ 0 ∈ ℝ
49		simp1l 1198	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
50		ltnle 11314	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ ¬ 𝐴 ≤ 0))
51	48, 49, 50	sylancr 587	. . . . . . 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 13051	. . . . . . . . . 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 26637	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
56	3, 54, 55	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) → (𝐴↑_𝑐𝐶) ∈ ℝ⁺)
57		rpgt0 13021	. . . . . . . . 9 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → 0 < (𝐴↑_𝑐𝐶))
58		rpre 13017	. . . . . . . . . 10 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → (𝐴↑_𝑐𝐶) ∈ ℝ)
59		ltnle 11314	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐴↑_𝑐𝐶) ∈ ℝ) → (0 < (𝐴↑_𝑐𝐶) ↔ ¬ (𝐴↑_𝑐𝐶) ≤ 0))
60	48, 58, 59	sylancr 587	. . . . . . . . 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 11263	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
64	9	rpne0d 13056	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
65		0cxp 26627	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (0↑_𝑐𝐶) = 0)
66	63, 64, 65	syl2anc 584	. . . . . . . . 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 5131	. . . . . . 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 5123	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ≤ 0 ↔ 𝐴 ≤ 𝐵))
72		oveq1 7412	. . . . . . 7 ⊢ (0 = 𝐵 → (0↑_𝑐𝐶) = (𝐵↑_𝑐𝐶))
73	72	breq2d 5131	. . . . . 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 1201	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 0 ≤ 𝐵)
78		leloe 11321	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
79	48, 5, 78	sylancr 587	. . . . 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 960	. 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 1228	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 ≤ 𝐵)
85	83, 84	eqbrtrrd 5143	. . 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 7420	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (0↑_𝑐𝐶) = (𝐴↑_𝑐𝐶))
88	86, 87	eqtr3d 2772	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 = (𝐴↑_𝑐𝐶))
89		simpl2l 1227	. . . . 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 26644	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐶 ∈ ℝ) → 0 ≤ (𝐵↑_𝑐𝐶))
92	89, 84, 90, 91	syl3anc 1373	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → 0 ≤ (𝐵↑_𝑐𝐶))
93	88, 92	eqbrtrrd 5143	. . 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 1199	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 0 ≤ 𝐴)
96		leloe 11321	. . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
97	48, 49, 96	sylancr 587	. . 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 960	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 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 · cmul 11134 < clt 11269 ≤ cle 11270 ℝ⁺crp 13008 expce 16077 logclog 26515 ↑_𝑐ccxp 26516

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517 df-cxp 26518

This theorem is referenced by: cxplt2 26659 cxple2a 26660 cxple2d 26688

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