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Theorem modxai 12991

Description: Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)

**Hypotheses**
Ref	Expression
modxai.1	⊢ 𝑁 ∈ ℕ
modxai.2	⊢ 𝐴 ∈ ℕ
modxai.3	⊢ 𝐵 ∈ ℕ₀
modxai.4	⊢ 𝐷 ∈ ℤ
modxai.5	⊢ 𝐾 ∈ ℕ₀
modxai.6	⊢ 𝑀 ∈ ℕ₀
modxai.7	⊢ 𝐶 ∈ ℕ₀
modxai.8	⊢ 𝐿 ∈ ℕ₀
modxai.11	⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁)
modxai.12	⊢ ((𝐴↑𝐶) mod 𝑁) = (𝐿 mod 𝑁)
modxai.9	⊢ (𝐵 + 𝐶) = 𝐸
modxai.10	⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿)

**Assertion**
Ref	Expression
modxai	⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)

**Proof of Theorem modxai**
Step	Hyp	Ref	Expression
1		modxai.9	. . . . 5 ⊢ (𝐵 + 𝐶) = 𝐸
2	1	oveq2i 6029	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = (𝐴↑𝐸)
3		modxai.2	. . . . . 6 ⊢ 𝐴 ∈ ℕ
4	3	nncni 9153	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		modxai.3	. . . . 5 ⊢ 𝐵 ∈ ℕ₀
6		modxai.7	. . . . 5 ⊢ 𝐶 ∈ ℕ₀
7		expadd 10844	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶)))
8	4, 5, 6, 7	mp3an 1373	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶))
9	2, 8	eqtr3i 2254	. . 3 ⊢ (𝐴↑𝐸) = ((𝐴↑𝐵) · (𝐴↑𝐶))
10	9	oveq1i 6028	. 2 ⊢ ((𝐴↑𝐸) mod 𝑁) = (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁)
11		nnexpcl 10815	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ₀) → (𝐴↑𝐵) ∈ ℕ)
12	3, 5, 11	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐵) ∈ ℕ
13	12	nnzi 9500	. . . . . . 7 ⊢ (𝐴↑𝐵) ∈ ℤ
14	13	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐵) ∈ ℤ)
15		modxai.5	. . . . . . . 8 ⊢ 𝐾 ∈ ℕ₀
16	15	nn0zi 9501	. . . . . . 7 ⊢ 𝐾 ∈ ℤ
17	16	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐾 ∈ ℤ)
18		nnexpcl 10815	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑𝐶) ∈ ℕ)
19	3, 6, 18	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐶) ∈ ℕ
20	19	nnzi 9500	. . . . . . 7 ⊢ (𝐴↑𝐶) ∈ ℤ
21	20	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐶) ∈ ℤ)
22		modxai.8	. . . . . . . 8 ⊢ 𝐿 ∈ ℕ₀
23	22	nn0zi 9501	. . . . . . 7 ⊢ 𝐿 ∈ ℤ
24	23	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐿 ∈ ℤ)
25		modxai.1	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
26		nnq 9867	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
27	25, 26	mp1i 10	. . . . . 6 ⊢ (⊤ → 𝑁 ∈ ℚ)
28		nnrp 9898	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
29	25, 28	mp1i 10	. . . . . . 7 ⊢ (⊤ → 𝑁 ∈ ℝ⁺)
30	29	rpgt0d 9934	. . . . . 6 ⊢ (⊤ → 0 < 𝑁)
31		modxai.11	. . . . . . 7 ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁)
32	31	a1i 9	. . . . . 6 ⊢ (⊤ → ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁))
33		modxai.12	. . . . . . 7 ⊢ ((𝐴↑𝐶) mod 𝑁) = (𝐿 mod 𝑁)
34	33	a1i 9	. . . . . 6 ⊢ (⊤ → ((𝐴↑𝐶) mod 𝑁) = (𝐿 mod 𝑁))
35	14, 17, 21, 24, 27, 30, 32, 34	modqmul12d 10641	. . . . 5 ⊢ (⊤ → (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁))
36	35	mptru 1406	. . . 4 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁)
37		modxai.10	. . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿)
38		modxai.4	. . . . . . . . 9 ⊢ 𝐷 ∈ ℤ
39		zcn 9484	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ)
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ 𝐷 ∈ ℂ
41	25	nncni 9153	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
42	40, 41	mulcli 8184	. . . . . . 7 ⊢ (𝐷 · 𝑁) ∈ ℂ
43		modxai.6	. . . . . . . 8 ⊢ 𝑀 ∈ ℕ₀
44	43	nn0cni 9414	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
45	42, 44	addcomi 8323	. . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝑀 + (𝐷 · 𝑁))
46	37, 45	eqtr3i 2254	. . . . 5 ⊢ (𝐾 · 𝐿) = (𝑀 + (𝐷 · 𝑁))
47	46	oveq1i 6028	. . . 4 ⊢ ((𝐾 · 𝐿) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
48	36, 47	eqtri 2252	. . 3 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
49		nn0z 9499	. . . . 5 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
50		zq 9860	. . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℚ)
51	43, 49, 50	mp2b 8	. . . 4 ⊢ 𝑀 ∈ ℚ
52	25, 26	ax-mp 5	. . . 4 ⊢ 𝑁 ∈ ℚ
53	30	mptru 1406	. . . 4 ⊢ 0 < 𝑁
54		modqcyc 10622	. . . 4 ⊢ (((𝑀 ∈ ℚ ∧ 𝐷 ∈ ℤ) ∧ (𝑁 ∈ ℚ ∧ 0 < 𝑁)) → ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁))
55	51, 38, 52, 53, 54	mp4an 427	. . 3 ⊢ ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁)
56	48, 55	eqtri 2252	. 2 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = (𝑀 mod 𝑁)
57	10, 56	eqtri 2252	1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Colors of variables: wff set class

Syntax hints: = wceq 1397 ⊤wtru 1398 ∈ wcel 2202 class class class wbr 4088 (class class class)co 6018 ℂcc 8030 0cc0 8032 + caddc 8035 · cmul 8037 < clt 8214 ℕcn 9143 ℕ₀cn0 9402 ℤcz 9479 ℚcq 9853 ℝ⁺crp 9888 mod cmo 10585 ↑cexp 10801

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802

This theorem is referenced by: mod2xi 12992 modxp1i 12993

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