mulexpzap - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mulexpzap		GIF version

Theorem mulexpzap 10326

Description: Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)

**Assertion**
Ref	Expression
mulexpzap	⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))

**Proof of Theorem mulexpzap**
Step	Hyp	Ref	Expression
1		elznn0nn 9061	. . 3 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
2		simpl 108	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 𝐴 ∈ ℂ)
3		simpl 108	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ∈ ℂ)
4	2, 3	anim12i 336	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
5		mulexp 10325	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
6	5	3expa 1181	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ₀) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
7	4, 6	sylan 281	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ 𝑁 ∈ ℕ₀) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
8		simplll 522	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → 𝐴 ∈ ℂ)
9		simplrl 524	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → 𝐵 ∈ ℂ)
10	8, 9	mulcld 7779	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐴 · 𝐵) ∈ ℂ)
11		simpllr 523	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → 𝐴 # 0)
12		simplrr 525	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → 𝐵 # 0)
13	8, 9, 11, 12	mulap0d 8412	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐴 · 𝐵) # 0)
14		recn 7746	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
15	14	ad2antrl 481	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → 𝑁 ∈ ℂ)
16		nnnn0 8977	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
17	16	ad2antll 482	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → -𝑁 ∈ ℕ₀)
18		expineg2 10295	. . . . . 6 ⊢ ((((𝐴 · 𝐵) ∈ ℂ ∧ (𝐴 · 𝐵) # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀)) → ((𝐴 · 𝐵)↑𝑁) = (1 / ((𝐴 · 𝐵)↑-𝑁)))
19	10, 13, 15, 17, 18	syl22anc 1217	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → ((𝐴 · 𝐵)↑𝑁) = (1 / ((𝐴 · 𝐵)↑-𝑁)))
20		expineg2 10295	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀)) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁)))
21	8, 11, 15, 17, 20	syl22anc 1217	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁)))
22		expineg2 10295	. . . . . . . 8 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀)) → (𝐵↑𝑁) = (1 / (𝐵↑-𝑁)))
23	9, 12, 15, 17, 22	syl22anc 1217	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐵↑𝑁) = (1 / (𝐵↑-𝑁)))
24	21, 23	oveq12d 5785	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → ((𝐴↑𝑁) · (𝐵↑𝑁)) = ((1 / (𝐴↑-𝑁)) · (1 / (𝐵↑-𝑁))))
25		mulexp 10325	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ -𝑁 ∈ ℕ₀) → ((𝐴 · 𝐵)↑-𝑁) = ((𝐴↑-𝑁) · (𝐵↑-𝑁)))
26	8, 9, 17, 25	syl3anc 1216	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → ((𝐴 · 𝐵)↑-𝑁) = ((𝐴↑-𝑁) · (𝐵↑-𝑁)))
27	26	oveq2d 5783	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (1 / ((𝐴 · 𝐵)↑-𝑁)) = (1 / ((𝐴↑-𝑁) · (𝐵↑-𝑁))))
28		1t1e1 8865	. . . . . . . . 9 ⊢ (1 · 1) = 1
29	28	oveq1i 5777	. . . . . . . 8 ⊢ ((1 · 1) / ((𝐴↑-𝑁) · (𝐵↑-𝑁))) = (1 / ((𝐴↑-𝑁) · (𝐵↑-𝑁)))
30	27, 29	syl6eqr 2188	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (1 / ((𝐴 · 𝐵)↑-𝑁)) = ((1 · 1) / ((𝐴↑-𝑁) · (𝐵↑-𝑁))))
31		expcl 10304	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ₀) → (𝐴↑-𝑁) ∈ ℂ)
32	8, 17, 31	syl2anc 408	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐴↑-𝑁) ∈ ℂ)
33		nnz 9066	. . . . . . . . . 10 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
34	33	ad2antll 482	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → -𝑁 ∈ ℤ)
35		expap0i 10318	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ -𝑁 ∈ ℤ) → (𝐴↑-𝑁) # 0)
36	8, 11, 34, 35	syl3anc 1216	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐴↑-𝑁) # 0)
37		expcl 10304	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ -𝑁 ∈ ℕ₀) → (𝐵↑-𝑁) ∈ ℂ)
38	9, 17, 37	syl2anc 408	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐵↑-𝑁) ∈ ℂ)
39		expap0i 10318	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 # 0 ∧ -𝑁 ∈ ℤ) → (𝐵↑-𝑁) # 0)
40	9, 12, 34, 39	syl3anc 1216	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝐵↑-𝑁) # 0)
41		ax-1cn 7706	. . . . . . . . 9 ⊢ 1 ∈ ℂ
42		divmuldivap 8465	. . . . . . . . 9 ⊢ (((1 ∈ ℂ ∧ 1 ∈ ℂ) ∧ (((𝐴↑-𝑁) ∈ ℂ ∧ (𝐴↑-𝑁) # 0) ∧ ((𝐵↑-𝑁) ∈ ℂ ∧ (𝐵↑-𝑁) # 0))) → ((1 / (𝐴↑-𝑁)) · (1 / (𝐵↑-𝑁))) = ((1 · 1) / ((𝐴↑-𝑁) · (𝐵↑-𝑁))))
43	41, 41, 42	mpanl12 432	. . . . . . . 8 ⊢ ((((𝐴↑-𝑁) ∈ ℂ ∧ (𝐴↑-𝑁) # 0) ∧ ((𝐵↑-𝑁) ∈ ℂ ∧ (𝐵↑-𝑁) # 0)) → ((1 / (𝐴↑-𝑁)) · (1 / (𝐵↑-𝑁))) = ((1 · 1) / ((𝐴↑-𝑁) · (𝐵↑-𝑁))))
44	32, 36, 38, 40, 43	syl22anc 1217	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → ((1 / (𝐴↑-𝑁)) · (1 / (𝐵↑-𝑁))) = ((1 · 1) / ((𝐴↑-𝑁) · (𝐵↑-𝑁))))
45	30, 44	eqtr4d 2173	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (1 / ((𝐴 · 𝐵)↑-𝑁)) = ((1 / (𝐴↑-𝑁)) · (1 / (𝐵↑-𝑁))))
46	24, 45	eqtr4d 2173	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → ((𝐴↑𝑁) · (𝐵↑𝑁)) = (1 / ((𝐴 · 𝐵)↑-𝑁)))
47	19, 46	eqtr4d 2173	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
48	7, 47	jaodan 786	. . 3 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
49	1, 48	sylan2b 285	. 2 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
50	49	3impa 1176	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∨ wo 697 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 ℝcr 7612 0cc0 7613 1c1 7614 · cmul 7618 -cneg 7927 # cap 8336 / cdiv 8425 ℕcn 8713 ℕ₀cn0 8970 ℤcz 9047 ↑cexp 10285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 df-exp 10286

This theorem is referenced by: exprecap 10327

W3C validator