exp1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > exp1		Structured version Visualization version GIF version

Theorem exp1 14058

Description: Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)

**Assertion**
Ref	Expression
exp1	⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴)

**Proof of Theorem exp1**
Step	Hyp	Ref	Expression
1		1nn 12247	. . . 4 ⊢ 1 ∈ ℕ
2		expnnval 14055	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℕ) → (𝐴↑1) = (seq1( · , (ℕ × {𝐴}))‘1))
3	1, 2	mpan2 690	. . 3 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = (seq1( · , (ℕ × {𝐴}))‘1))
4		1z 12616	. . . 4 ⊢ 1 ∈ ℤ
5		seq1 14005	. . . 4 ⊢ (1 ∈ ℤ → (seq1( · , (ℕ × {𝐴}))‘1) = ((ℕ × {𝐴})‘1))
6	4, 5	ax-mp 5	. . 3 ⊢ (seq1( · , (ℕ × {𝐴}))‘1) = ((ℕ × {𝐴})‘1)
7	3, 6	eqtrdi 2784	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = ((ℕ × {𝐴})‘1))
8		fvconst2g 7208	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℕ) → ((ℕ × {𝐴})‘1) = 𝐴)
9	1, 8	mpan2 690	. 2 ⊢ (𝐴 ∈ ℂ → ((ℕ × {𝐴})‘1) = 𝐴)
10	7, 9	eqtrd 2768	1 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {csn 4624 × cxp 5670 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 1c1 11133 · cmul 11137 ℕcn 12236 ℤcz 12582 seqcseq 13992 ↑cexp 14052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-seq 13993 df-exp 14053

This theorem is referenced by: expp1 14059 expn1 14062 expcllem 14063 expeq0 14083 expp1z 14102 expm1 14103 sqval 14105 exp1d 14131 expmordi 14157 expnbnd 14220 digit1 14225 faclbnd4lem1 14278 climcndslem1 15821 climcndslem2 15822 geoisum1 15851 bpoly1 16021 ef4p 16083 efgt1p2 16084 efgt1p 16085 rpnnen2lem3 16186 modxp1i 17032 numexp1 17039 psgnpmtr 19458 lt6abl 19843 cphipval 25164 iblcnlem1 25710 itgcnlem 25712 dvexp 25878 dveflem 25904 plyid 26136 coeidp 26191 dgrid 26192 cxp1 26598 1cubrlem 26766 1cubr 26767 log2ublem3 26873 basellem5 27010 perfectlem2 27156 logdivsum 27459 log2sumbnd 27470 ipval2 30510 0dp2dp 32626 subfacval2 34791 dvasin 37171 areacirclem1 37175 1t10e1p1e11 46684 fmtnoge3 46864 fmtno0 46874 fmtno1 46875 lighneallem2 46940 lighneallem3 46941 41prothprmlem2 46952 perfectALTVlem2 47056 8exp8mod9 47070 tgblthelfgott 47149 exple2lt6 47422 pw2m1lepw2m1 47582 logbpw2m1 47634 nnpw2pmod 47650

W3C validator

	
		OSZAR »