cxpge0 - Metamath Proof Explorer

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Theorem cxpge0 26725

Description: Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)

**Assertion**
Ref	Expression
cxpge0	⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → 0 ≤ (𝐴↑_𝑐𝐵))

**Proof of Theorem cxpge0**
Step	Hyp	Ref	Expression
1		0re 11180	. . . . . 6 ⊢ 0 ∈ ℝ
2		leloe 11266	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
3	1, 2	mpan 700	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
4	3	adantr 484	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
5		elrp 12992	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
6		rpcxpcl 26718	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴↑_𝑐𝐵) ∈ ℝ⁺)
7	6	rpge0d 13038	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 0 ≤ (𝐴↑_𝑐𝐵))
8	7	ex 416	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (𝐵 ∈ ℝ → 0 ≤ (𝐴↑_𝑐𝐵)))
9	5, 8	sylbir 237	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐵 ∈ ℝ → 0 ≤ (𝐴↑_𝑐𝐵)))
10	9	impancom 455	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 → 0 ≤ (𝐴↑_𝑐𝐵)))
11		0le1 11707	. . . . . . . . . 10 ⊢ 0 ≤ 1
12		0cn 11168	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
13		cxp0 26712	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → (0↑_𝑐0) = 1)
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (0↑_𝑐0) = 1
15	11, 14	breqtrri 5126	. . . . . . . . 9 ⊢ 0 ≤ (0↑_𝑐0)
16		simpr 488	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 = 0) → 𝐵 = 0)
17	16	oveq2d 7408	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 = 0) → (0↑_𝑐𝐵) = (0↑_𝑐0))
18	15, 17	breqtrrid 5137	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 = 0) → 0 ≤ (0↑_𝑐𝐵))
19		0le0 12316	. . . . . . . . 9 ⊢ 0 ≤ 0
20		recn 11160	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
21		0cxp 26708	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
22	20, 21	sylan 589	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
23	19, 22	breqtrrid 5137	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → 0 ≤ (0↑_𝑐𝐵))
24	18, 23	pm2.61dane 3043	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 0 ≤ (0↑_𝑐𝐵))
25	24	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ (0↑_𝑐𝐵))
26		oveq1 7399	. . . . . . 7 ⊢ (0 = 𝐴 → (0↑_𝑐𝐵) = (𝐴↑_𝑐𝐵))
27	26	breq2d 5111	. . . . . 6 ⊢ (0 = 𝐴 → (0 ≤ (0↑_𝑐𝐵) ↔ 0 ≤ (𝐴↑_𝑐𝐵)))
28	25, 27	syl5ibcom 247	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 = 𝐴 → 0 ≤ (𝐴↑_𝑐𝐵)))
29	10, 28	jaod 870	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∨ 0 = 𝐴) → 0 ≤ (𝐴↑_𝑐𝐵)))
30	4, 29	sylbid 242	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐴 → 0 ≤ (𝐴↑_𝑐𝐵)))
31	30	3impia 1129	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑_𝑐𝐵))
32	31	3com23 1138	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → 0 ≤ (𝐴↑_𝑐𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 class class class wbr 5099 (class class class)co 7392 ℂcc 11068 ℝcr 11069 0cc0 11070 1c1 11071 < clt 11213 ≤ cle 11214 ℝ⁺crp 12990 ↑_𝑐ccxp 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ioc 13351 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-fac 14284 df-bc 14313 df-hash 14341 df-shft 15077 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-rlim 15499 df-sum 15697 df-ef 16080 df-sin 16082 df-cos 16083 df-pi 16085 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-fbas 21401 df-fg 21402 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-lp 23176 df-perf 23177 df-cn 23267 df-cnp 23268 df-haus 23355 df-tx 23602 df-hmeo 23795 df-fil 23886 df-fm 23978 df-flim 23979 df-flf 23980 df-xms 24360 df-ms 24361 df-tms 24362 df-cncf 24920 df-limc 25908 df-dv 25909 df-log 26598 df-cxp 26599

This theorem is referenced by: abscxp2 26735 cxple2 26739 cxpge0d 26766 cxpaddlelem 26793

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