modxai - Intuitionistic Logic Explorer

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Theorem modxai 12905

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 5985	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = (𝐴↑𝐸)
3		modxai.2	. . . . . 6 ⊢ 𝐴 ∈ ℕ
4	3	nncni 9088	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		modxai.3	. . . . 5 ⊢ 𝐵 ∈ ℕ₀
6		modxai.7	. . . . 5 ⊢ 𝐶 ∈ ℕ₀
7		expadd 10770	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶)))
8	4, 5, 6, 7	mp3an 1352	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶))
9	2, 8	eqtr3i 2232	. . 3 ⊢ (𝐴↑𝐸) = ((𝐴↑𝐵) · (𝐴↑𝐶))
10	9	oveq1i 5984	. 2 ⊢ ((𝐴↑𝐸) mod 𝑁) = (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁)
11		nnexpcl 10741	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ₀) → (𝐴↑𝐵) ∈ ℕ)
12	3, 5, 11	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐵) ∈ ℕ
13	12	nnzi 9435	. . . . . . 7 ⊢ (𝐴↑𝐵) ∈ ℤ
14	13	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐵) ∈ ℤ)
15		modxai.5	. . . . . . . 8 ⊢ 𝐾 ∈ ℕ₀
16	15	nn0zi 9436	. . . . . . 7 ⊢ 𝐾 ∈ ℤ
17	16	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐾 ∈ ℤ)
18		nnexpcl 10741	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑𝐶) ∈ ℕ)
19	3, 6, 18	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐶) ∈ ℕ
20	19	nnzi 9435	. . . . . . 7 ⊢ (𝐴↑𝐶) ∈ ℤ
21	20	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐶) ∈ ℤ)
22		modxai.8	. . . . . . . 8 ⊢ 𝐿 ∈ ℕ₀
23	22	nn0zi 9436	. . . . . . 7 ⊢ 𝐿 ∈ ℤ
24	23	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐿 ∈ ℤ)
25		modxai.1	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
26		nnq 9796	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
27	25, 26	mp1i 10	. . . . . 6 ⊢ (⊤ → 𝑁 ∈ ℚ)
28		nnrp 9827	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
29	25, 28	mp1i 10	. . . . . . 7 ⊢ (⊤ → 𝑁 ∈ ℝ⁺)
30	29	rpgt0d 9863	. . . . . 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 10567	. . . . 5 ⊢ (⊤ → (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁))
36	35	mptru 1384	. . . 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 9419	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ)
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ 𝐷 ∈ ℂ
41	25	nncni 9088	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
42	40, 41	mulcli 8119	. . . . . . 7 ⊢ (𝐷 · 𝑁) ∈ ℂ
43		modxai.6	. . . . . . . 8 ⊢ 𝑀 ∈ ℕ₀
44	43	nn0cni 9349	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
45	42, 44	addcomi 8258	. . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝑀 + (𝐷 · 𝑁))
46	37, 45	eqtr3i 2232	. . . . 5 ⊢ (𝐾 · 𝐿) = (𝑀 + (𝐷 · 𝑁))
47	46	oveq1i 5984	. . . 4 ⊢ ((𝐾 · 𝐿) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
48	36, 47	eqtri 2230	. . 3 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
49		nn0z 9434	. . . . 5 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
50		zq 9789	. . . . 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 1384	. . . 4 ⊢ 0 < 𝑁
54		modqcyc 10548	. . . 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 2230	. 2 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = (𝑀 mod 𝑁)
57	10, 56	eqtri 2230	1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Colors of variables: wff set class

Syntax hints: = wceq 1375 ⊤wtru 1376 ∈ wcel 2180 class class class wbr 4062 (class class class)co 5974 ℂcc 7965 0cc0 7967 + caddc 7970 · cmul 7972 < clt 8149 ℕcn 9078 ℕ₀cn0 9337 ℤcz 9414 ℚcq 9782 ℝ⁺crp 9817 mod cmo 10511 ↑cexp 10727

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-frec 6507 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-fl 10457 df-mod 10512 df-seqfrec 10637 df-exp 10728

This theorem is referenced by: mod2xi 12906 modxp1i 12907

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