cxpaddlelem - Metamath Proof Explorer

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Theorem cxpaddlelem 26495

Description: Lemma for cxpaddle 26496. (Contributed by Mario Carneiro, 2-Aug-2014.)

**Hypotheses**
Ref	Expression
cxpaddlelem.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
cxpaddlelem.2	⊢ (𝜑 → 0 ≤ 𝐴)
cxpaddlelem.3	⊢ (𝜑 → 𝐴 ≤ 1)
cxpaddlelem.4	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
cxpaddlelem.5	⊢ (𝜑 → 𝐵 ≤ 1)

**Assertion**
Ref	Expression
cxpaddlelem	⊢ (𝜑 → 𝐴 ≤ (𝐴↑_𝑐𝐵))

**Proof of Theorem cxpaddlelem**
Step	Hyp	Ref	Expression
1		cxpaddlelem.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		cxpaddlelem.2	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐴)
3		1re 11218	. . . . . . 7 ⊢ 1 ∈ ℝ
4		cxpaddlelem.4	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
5	4	rpred 13020	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6		resubcl 11528	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1 − 𝐵) ∈ ℝ)
7	3, 5, 6	sylancr 585	. . . . . 6 ⊢ (𝜑 → (1 − 𝐵) ∈ ℝ)
8	1, 2, 7	recxpcld 26467	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐(1 − 𝐵)) ∈ ℝ)
9	8	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑_𝑐(1 − 𝐵)) ∈ ℝ)
10		1red 11219	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → 1 ∈ ℝ)
11		recxpcl 26419	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → (𝐴↑_𝑐𝐵) ∈ ℝ)
12		cxpge0 26427	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → 0 ≤ (𝐴↑_𝑐𝐵))
13	11, 12	jca 510	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝐴↑_𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑_𝑐𝐵)))
14	1, 2, 5, 13	syl3anc 1369	. . . . 5 ⊢ (𝜑 → ((𝐴↑_𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑_𝑐𝐵)))
15	14	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑_𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑_𝑐𝐵)))
16		cxpaddlelem.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ 1)
17	16	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 𝐴 ≤ 1)
18	1	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 𝐴 ∈ ℝ)
19	2	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 0 ≤ 𝐴)
20		1red 11219	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 1 ∈ ℝ)
21		0le1 11741	. . . . . . . . 9 ⊢ 0 ≤ 1
22	21	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 0 ≤ 1)
23		difrp 13016	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐵 < 1 ↔ (1 − 𝐵) ∈ ℝ⁺))
24	5, 3, 23	sylancl 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 < 1 ↔ (1 − 𝐵) ∈ ℝ⁺))
25	24	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐵 < 1 ↔ (1 − 𝐵) ∈ ℝ⁺))
26	25	biimpa 475	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (1 − 𝐵) ∈ ℝ⁺)
27	18, 19, 20, 22, 26	cxple2d 26471	. . . . . . 7 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴 ≤ 1 ↔ (𝐴↑_𝑐(1 − 𝐵)) ≤ (1↑_𝑐(1 − 𝐵))))
28	17, 27	mpbid 231	. . . . . 6 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴↑_𝑐(1 − 𝐵)) ≤ (1↑_𝑐(1 − 𝐵)))
29	7	recnd 11246	. . . . . . . 8 ⊢ (𝜑 → (1 − 𝐵) ∈ ℂ)
30	29	1cxpd 26451	. . . . . . 7 ⊢ (𝜑 → (1↑_𝑐(1 − 𝐵)) = 1)
31	30	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (1↑_𝑐(1 − 𝐵)) = 1)
32	28, 31	breqtrd 5173	. . . . 5 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴↑_𝑐(1 − 𝐵)) ≤ 1)
33		simpr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → 𝐵 = 1)
34	33	oveq2d 7427	. . . . . . . . 9 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (1 − 𝐵) = (1 − 1))
35		1m1e0 12288	. . . . . . . . 9 ⊢ (1 − 1) = 0
36	34, 35	eqtrdi 2786	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (1 − 𝐵) = 0)
37	36	oveq2d 7427	. . . . . . 7 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑_𝑐(1 − 𝐵)) = (𝐴↑_𝑐0))
38	1	recnd 11246	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
39	38	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → 𝐴 ∈ ℂ)
40	39	cxp0d 26449	. . . . . . 7 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑_𝑐0) = 1)
41	37, 40	eqtrd 2770	. . . . . 6 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑_𝑐(1 − 𝐵)) = 1)
42		1le1 11846	. . . . . 6 ⊢ 1 ≤ 1
43	41, 42	eqbrtrdi 5186	. . . . 5 ⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑_𝑐(1 − 𝐵)) ≤ 1)
44		cxpaddlelem.5	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 1)
45		leloe 11304	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐵 ≤ 1 ↔ (𝐵 < 1 ∨ 𝐵 = 1)))
46	5, 3, 45	sylancl 584	. . . . . . 7 ⊢ (𝜑 → (𝐵 ≤ 1 ↔ (𝐵 < 1 ∨ 𝐵 = 1)))
47	44, 46	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐵 < 1 ∨ 𝐵 = 1))
48	47	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐵 < 1 ∨ 𝐵 = 1))
49	32, 43, 48	mpjaodan 955	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑_𝑐(1 − 𝐵)) ≤ 1)
50		lemul1a 12072	. . . 4 ⊢ ((((𝐴↑_𝑐(1 − 𝐵)) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝐴↑_𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑_𝑐𝐵))) ∧ (𝐴↑_𝑐(1 − 𝐵)) ≤ 1) → ((𝐴↑_𝑐(1 − 𝐵)) · (𝐴↑_𝑐𝐵)) ≤ (1 · (𝐴↑_𝑐𝐵)))
51	9, 10, 15, 49, 50	syl31anc 1371	. . 3 ⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑_𝑐(1 − 𝐵)) · (𝐴↑_𝑐𝐵)) ≤ (1 · (𝐴↑_𝑐𝐵)))
52		ax-1cn 11170	. . . . . . 7 ⊢ 1 ∈ ℂ
53	5	recnd 11246	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
54		npcan 11473	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 − 𝐵) + 𝐵) = 1)
55	52, 53, 54	sylancr 585	. . . . . 6 ⊢ (𝜑 → ((1 − 𝐵) + 𝐵) = 1)
56	55	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → ((1 − 𝐵) + 𝐵) = 1)
57	56	oveq2d 7427	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑_𝑐((1 − 𝐵) + 𝐵)) = (𝐴↑_𝑐1))
58	38	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℂ)
59	1	anim1i 613	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
60		elrp 12980	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
61	59, 60	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ⁺)
62	61	rpne0d 13025	. . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ≠ 0)
63	29	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → (1 − 𝐵) ∈ ℂ)
64	53	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐵 ∈ ℂ)
65	58, 62, 63, 64	cxpaddd 26461	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑_𝑐((1 − 𝐵) + 𝐵)) = ((𝐴↑_𝑐(1 − 𝐵)) · (𝐴↑_𝑐𝐵)))
66	38	cxp1d 26450	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐1) = 𝐴)
67	66	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑_𝑐1) = 𝐴)
68	57, 65, 67	3eqtr3d 2778	. . 3 ⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑_𝑐(1 − 𝐵)) · (𝐴↑_𝑐𝐵)) = 𝐴)
69	38, 53	cxpcld 26452	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐𝐵) ∈ ℂ)
70	69	mullidd 11236	. . . 4 ⊢ (𝜑 → (1 · (𝐴↑_𝑐𝐵)) = (𝐴↑_𝑐𝐵))
71	70	adantr 479	. . 3 ⊢ ((𝜑 ∧ 0 < 𝐴) → (1 · (𝐴↑_𝑐𝐵)) = (𝐴↑_𝑐𝐵))
72	51, 68, 71	3brtr3d 5178	. 2 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ≤ (𝐴↑_𝑐𝐵))
73	1, 2, 5	cxpge0d 26468	. . . 4 ⊢ (𝜑 → 0 ≤ (𝐴↑_𝑐𝐵))
74		breq1 5150	. . . 4 ⊢ (0 = 𝐴 → (0 ≤ (𝐴↑_𝑐𝐵) ↔ 𝐴 ≤ (𝐴↑_𝑐𝐵)))
75	73, 74	syl5ibcom 244	. . 3 ⊢ (𝜑 → (0 = 𝐴 → 𝐴 ≤ (𝐴↑_𝑐𝐵)))
76	75	imp 405	. 2 ⊢ ((𝜑 ∧ 0 = 𝐴) → 𝐴 ≤ (𝐴↑_𝑐𝐵))
77		0re 11220	. . . 4 ⊢ 0 ∈ ℝ
78		leloe 11304	. . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
79	77, 1, 78	sylancr 585	. . 3 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
80	2, 79	mpbid 231	. 2 ⊢ (𝜑 → (0 < 𝐴 ∨ 0 = 𝐴))
81	72, 76, 80	mpjaodan 955	1 ⊢ (𝜑 → 𝐴 ≤ (𝐴↑_𝑐𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 ℝ⁺crp 12978 ↑_𝑐ccxp 26300

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-cxp 26302

This theorem is referenced by: cxpaddle 26496

W3C validator