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Theorem cxple2 27058

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 13191	. . . . 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 13191	. . . 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 13194	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
13		relogcl 26936	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
14	13	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐴) ∈ ℝ)
15	12, 14	remulcld 11369	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐴)) ∈ ℝ)
16		relogcl 26936	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
17	16	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐵) ∈ ℝ)
18	12, 17	remulcld 11369	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐵)) ∈ ℝ)
19		efle 16318	. . . . . 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 16318	. . . . . . 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 13238	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵))))
24		reeflog 26941	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(log‘𝐴)) = 𝐴)
25	24	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐴)) = 𝐴)
26		reeflog 26941	. . . . . . . 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 5119	. . . . . 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 13159	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
31	30	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
32	31	recnd 11367	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
33		rpne0 13167	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
34	33	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≠ 0)
35	12	recnd 11367	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
36		cxpef 27026	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
37	32, 34, 35, 36	syl3anc 1399	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
38		rpre 13159	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
39	38	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
40	39	recnd 11367	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
41		rpne0 13167	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ≠ 0)
42	41	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ≠ 0)
43		cxpef 27026	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
44	40, 42, 35, 43	syl3anc 1399	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
45	37, 44	breq12d 5119	. . . . 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 1399	. . 3 ⊢ (((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 < 𝐴) ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴↑_𝑐𝐶) ≤ (𝐵↑_𝑐𝐶)))
48		0re 11340	. . . . . . . 8 ⊢ 0 ∈ ℝ
49		simp1l 1216	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
50		ltnle 11419	. . . . . . . 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 13194	. . . . . . . . . 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 27037	. . . . . . . . 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 13163	. . . . . . . . 9 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → 0 < (𝐴↑_𝑐𝐶))
58		rpre 13159	. . . . . . . . . 10 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → (𝐴↑_𝑐𝐶) ∈ ℝ)
59		ltnle 11419	. . . . . . . . . 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 11367	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
64	9	rpne0d 13199	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
65		0cxp 27027	. . . . . . . . . 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 5118	. . . . . . 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 5110	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ≤ 0 ↔ 𝐴 ≤ 𝐵))
72		oveq1 7453	. . . . . . 7 ⊢ (0 = 𝐵 → (0↑_𝑐𝐶) = (𝐵↑_𝑐𝐶))
73	72	breq2d 5118	. . . . . 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 11426	. . . . . 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 7461	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (0↑_𝑐𝐶) = (𝐴↑_𝑐𝐶))
88	86, 87	eqtr3d 2800	. . . 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 27044	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐶 ∈ ℝ) → 0 ≤ (𝐵↑_𝑐𝐶))
92	89, 84, 90, 91	syl3anc 1399	. . . 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 11426	. . . 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 2147 ≠ wne 2959 class class class wbr 5106 ‘cfv 6561 (class class class)co 7446 ℂcc 11228 ℝcr 11229 0cc0 11230 · cmul 11235 < clt 11373 ≤ cle 11374 ℝ⁺crp 13150 expce 16259 logclog 26915 ↑_𝑐ccxp 26916

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308 ax-addf 11309

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-supp 8208 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-map 8880 df-pm 8881 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-fsupp 9385 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-q 13106 df-rp 13151 df-xneg 13271 df-xadd 13272 df-xmul 13273 df-ioo 13510 df-ioc 13511 df-ico 13512 df-icc 13513 df-fz 13670 df-fzo 13819 df-fl 13962 df-mod 14040 df-seq 14175 df-exp 14235 df-fac 14448 df-bc 14477 df-hash 14505 df-shft 15252 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-limsup 15670 df-clim 15687 df-rlim 15688 df-sum 15886 df-ef 16265 df-sin 16267 df-cos 16268 df-pi 16270 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-starv 17475 df-sca 17476 df-vsca 17477 df-ip 17478 df-tset 17479 df-ple 17480 df-ds 17482 df-unif 17483 df-hom 17484 df-cco 17485 df-rest 17625 df-topn 17626 df-0g 17644 df-gsum 17645 df-topgen 17646 df-pt 17647 df-prds 17650 df-xrs 17706 df-qtop 17711 df-imas 17712 df-xps 17714 df-mre 17788 df-mrc 17789 df-acs 17791 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-mulg 19310 df-cntz 19564 df-cmn 20029 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706 df-mopn 21707 df-fbas 21708 df-fg 21709 df-cnfld 21712 df-top 23245 df-topon 23262 df-topsp 23284 df-bases 23297 df-cld 23370 df-ntr 23371 df-cls 23372 df-nei 23449 df-lp 23487 df-perf 23488 df-cn 23578 df-cnp 23579 df-haus 23666 df-tx 23914 df-hmeo 24107 df-fil 24198 df-fm 24290 df-flim 24291 df-flf 24292 df-xms 24672 df-ms 24673 df-tms 24674 df-cncf 25232 df-limc 26219 df-dv 26220 df-log 26917 df-cxp 26918

This theorem is used by: cxplt2 27059 cxple2a 27060 cxple2d 27088

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