cxpmul2z - Metamath Proof Explorer

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Theorem cxpmul2z 25751

Description: Generalize cxpmul2 25749 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 12264	. . 3 ⊢ (𝐶 ∈ ℤ ↔ (𝐶 ∈ ℝ ∧ (𝐶 ∈ ℕ₀ ∨ -𝐶 ∈ ℕ₀)))
2		cxpmul2 25749	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶))
3	2	3expia 1119	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 ∈ ℕ₀ → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
4	3	ad4ant13 747	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ℕ₀ → (𝐴↑_𝑐(𝐵 · 𝐶)) = ((𝐴↑_𝑐𝐵)↑𝐶)))
5		simplll 771	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐴 ∈ ℂ)
6		simplr 765	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐵 ∈ ℂ)
7		simprr 769	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -𝐶 ∈ ℕ₀)
8		cxpmul2 25749	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ -𝐶 ∈ ℕ₀) → (𝐴↑_𝑐(𝐵 · -𝐶)) = ((𝐴↑_𝑐𝐵)↑-𝐶))
9	5, 6, 7, 8	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐(𝐵 · -𝐶)) = ((𝐴↑_𝑐𝐵)↑-𝐶))
10	9	oveq2d 7271	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (1 / (𝐴↑_𝑐(𝐵 · -𝐶))) = (1 / ((𝐴↑_𝑐𝐵)↑-𝐶)))
11		simprl 767	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐶 ∈ ℝ)
12	11	recnd 10934	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐶 ∈ ℂ)
13	6, 12	mulneg2d 11359	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐵 · -𝐶) = -(𝐵 · 𝐶))
14	13	negeqd 11145	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -(𝐵 · -𝐶) = --(𝐵 · 𝐶))
15	6, 12	mulcld 10926	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐵 · 𝐶) ∈ ℂ)
16	15	negnegd 11253	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → --(𝐵 · 𝐶) = (𝐵 · 𝐶))
17	14, 16	eqtrd 2778	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -(𝐵 · -𝐶) = (𝐵 · 𝐶))
18	17	oveq2d 7271	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐-(𝐵 · -𝐶)) = (𝐴↑_𝑐(𝐵 · 𝐶)))
19		simpllr 772	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → 𝐴 ≠ 0)
20	12	negcld 11249	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → -𝐶 ∈ ℂ)
21	6, 20	mulcld 10926	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐵 · -𝐶) ∈ ℂ)
22		cxpneg 25741	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (𝐵 · -𝐶) ∈ ℂ) → (𝐴↑_𝑐-(𝐵 · -𝐶)) = (1 / (𝐴↑_𝑐(𝐵 · -𝐶))))
23	5, 19, 21, 22	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐-(𝐵 · -𝐶)) = (1 / (𝐴↑_𝑐(𝐵 · -𝐶))))
24	18, 23	eqtr3d 2780	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐(𝐵 · 𝐶)) = (1 / (𝐴↑_𝑐(𝐵 · -𝐶))))
25		cxpcl 25734	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) ∈ ℂ)
26	25	ad4ant13 747	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → (𝐴↑_𝑐𝐵) ∈ ℂ)
27		expneg2 13719	. . . . . . . 8 ⊢ (((𝐴↑_𝑐𝐵) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ -𝐶 ∈ ℕ₀) → ((𝐴↑_𝑐𝐵)↑𝐶) = (1 / ((𝐴↑_𝑐𝐵)↑-𝐶)))
28	26, 12, 7, 27	syl3anc 1369	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℝ ∧ -𝐶 ∈ ℕ₀)) → ((𝐴↑_𝑐𝐵)↑𝐶) = (1 / ((𝐴↑_𝑐𝐵)↑-𝐶)))
29	10, 24, 28	3eqtr4d 2788	. . . . . 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 855	. . . 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	syl5bi 241	. 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 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 -cneg 11136 / cdiv 11562 ℕ₀cn0 12163 ℤcz 12249 ↑cexp 13710 ↑_𝑐ccxp 25616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618

This theorem is referenced by: cxpmul2zd 25776

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