cxpmul2z - Metamath Proof Explorer

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Theorem cxpmul2z 26576

Description: Generalize cxpmul2 26574 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)

**Assertion**
Ref	Expression
cxpmul2z	⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ)) → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶))

**Proof of Theorem cxpmul2z**
Step	Hyp	Ref	Expression
1		elznn0 12520	. . 3 ⊢ (𝐶 ∈ ℤ ↔ (𝐶 ∈ ℝ ∧ (𝐶 ∈ ℕ₀ ∨ -𝐶 ∈ ℕ₀)))
2		cxpmul2 26574	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶))
3	2	3expia 1121	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 ∈ ℕ₀ → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
4	3	ad4ant13 751	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ℕ₀ → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
5		simplll 774	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐴 ∈ ℂ)
6		simplr 768	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐵 ∈ ℂ)
7		simprr 772	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -𝐶 ∈ ℕ₀)
8		cxpmul2 26574	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ -𝐶 ∈ ℕ₀) → (𝐴↑_𝑐(𝐵 · -𝐶)) = ((𝐴↑_𝑐𝐵)↑-𝐶))
9	5, 6, 7, 8	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐(𝐵 · -𝐶)) = ((𝐴↑_𝑐𝐵)↑-𝐶))
10	9	oveq2d 7385	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (1 / (𝐴↑_𝑐(𝐵 · -𝐶))) = (1 / ((𝐴↑_𝑐𝐵)↑-𝐶)))
11		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐶 ∈ ℝ)
12	11	recnd 11178	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐶 ∈ ℂ)
13	6, 12	mulneg2d 11608	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐵 · -𝐶) = -(𝐵 · 𝐶))
14	13	negeqd 11391	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -(𝐵 · -𝐶) = --(𝐵 · 𝐶))
15	6, 12	mulcld 11170	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐵 · 𝐶) ∈ ℂ)
16	15	negnegd 11500	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → --(𝐵 · 𝐶) = (𝐵 · 𝐶))
17	14, 16	eqtrd 2764	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -(𝐵 · -𝐶) = (𝐵 · 𝐶))
18	17	oveq2d 7385	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐-(𝐵 · -𝐶)) = (𝐴↑_𝑐(𝐵 · 𝐶)))
19		simpllr 775	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐴 ≠ 0)
20	12	negcld 11496	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -𝐶 ∈ ℂ)
21	6, 20	mulcld 11170	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐵 · -𝐶) ∈ ℂ)
22		cxpneg 26566	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (𝐵 · -𝐶) ∈ ℂ) → (𝐴↑_𝑐-(𝐵 · -𝐶)) = (1 / (𝐴↑_𝑐(𝐵 · -𝐶))))
23	5, 19, 21, 22	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐-(𝐵 · -𝐶)) = (1 / (𝐴↑_𝑐(𝐵 · -𝐶))))
24	18, 23	eqtr3d 2766	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐(𝐵 · 𝐶)) = (1 / (𝐴↑_𝑐(𝐵 · -𝐶))))
25		cxpcl 26559	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) ∈ ℂ)
26	25	ad4ant13 751	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐𝐵) ∈ ℂ)
27		expneg2 14011	. . . . . . . 8 ⊢ (((𝐴↑_𝑐𝐵) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ -𝐶 ∈ ℕ₀) → ((𝐴↑_𝑐𝐵)↑𝐶) = (1 / ((𝐴↑_𝑐𝐵)↑-𝐶)))
28	26, 12, 7, 27	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → ((𝐴↑_𝑐𝐵)↑𝐶) = (1 / ((𝐴↑_𝑐𝐵)↑-𝐶)))
29	10, 24, 28	3eqtr4d 2774	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶))
30	29	expr 456	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ 𝐶 ∈ ℝ) → (-𝐶 ∈ ℕ₀ → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
31	4, 30	jaod 859	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ 𝐶 ∈ ℝ) → ((𝐶 ∈ ℕ₀ ∨ -𝐶 ∈ ℕ₀) → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
32	31	expimpd 453	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) → ((𝐶 ∈ ℝ ∧ (𝐶 ∈ ℕ₀ ∨ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
33	1, 32	biimtrid 242	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) → (𝐶 ∈ ℤ → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
34	33	impr 454	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ)) → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 -cneg 11382 / cdiv 11811 ℕ₀cn0 12418 ℤcz 12505 ↑cexp 14002 ↑_𝑐ccxp 26440

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-log 26441 df-cxp 26442

This theorem is referenced by: cxpmul2zd 26601

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