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Theorem cxple2 27031

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 13165	. . . . 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 13165	. . . 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 13168	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
13		relogcl 26909	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
14	13	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐴) ∈ ℝ)
15	12, 14	remulcld 11343	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐴)) ∈ ℝ)
16		relogcl 26909	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
17	16	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐵) ∈ ℝ)
18	12, 17	remulcld 11343	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶 · (log‘𝐵)) ∈ ℝ)
19		efle 16292	. . . . . 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 16292	. . . . . . 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 13212	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → ((log‘𝐴) ≤ (log‘𝐵) ↔ (𝐶 · (log‘𝐴)) ≤ (𝐶 · (log‘𝐵))))
24		reeflog 26914	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(log‘𝐴)) = 𝐴)
25	24	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (exp‘(log‘𝐴)) = 𝐴)
26		reeflog 26914	. . . . . . . 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 5115	. . . . . 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 13133	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
31	30	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
32	31	recnd 11341	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
33		rpne0 13141	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
34	33	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≠ 0)
35	12	recnd 11341	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
36		cxpef 26999	. . . . . . 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 13133	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
39	38	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
40	39	recnd 11341	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
41		rpne0 13141	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ≠ 0)
42	41	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐵 ≠ 0)
43		cxpef 26999	. . . . . . 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 5115	. . . . 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 11314	. . . . . . . 8 ⊢ 0 ∈ ℝ
49		simp1l 1216	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
50		ltnle 11393	. . . . . . . 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 13168	. . . . . . . . . 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 27010	. . . . . . . . 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 13137	. . . . . . . . 9 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → 0 < (𝐴↑_𝑐𝐶))
58		rpre 13133	. . . . . . . . . 10 ⊢ ((𝐴↑_𝑐𝐶) ∈ ℝ⁺ → (𝐴↑_𝑐𝐶) ∈ ℝ)
59		ltnle 11393	. . . . . . . . . 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 11341	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
64	9	rpne0d 13173	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
65		0cxp 27000	. . . . . . . . . 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 5114	. . . . . . 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 5106	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ≤ 0 ↔ 𝐴 ≤ 𝐵))
72		oveq1 7428	. . . . . . 7 ⊢ (0 = 𝐵 → (0↑_𝑐𝐶) = (𝐵↑_𝑐𝐶))
73	72	breq2d 5114	. . . . . 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 11400	. . . . . 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 5128	. . 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 7436	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) ∧ 0 = 𝐴) → (0↑_𝑐𝐶) = (𝐴↑_𝑐𝐶))
88	86, 87	eqtr3d 2797	. . . 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 27017	. . . . 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 5128	. . 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 11400	. . . 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 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 · cmul 11209 < clt 11347 ≤ cle 11348 ℝ⁺crp 13124 expce 16233 logclog 26888 ↑_𝑐ccxp 26889

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-pi 16244 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193 df-log 26890 df-cxp 26891

This theorem is used by: cxplt2 27032 cxple2a 27033 cxple2d 27061

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