modxai - Intuitionistic Logic Explorer

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Theorem modxai 12960

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 6021	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = (𝐴↑𝐸)
3		modxai.2	. . . . . 6 ⊢ 𝐴 ∈ ℕ
4	3	nncni 9136	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		modxai.3	. . . . 5 ⊢ 𝐵 ∈ ℕ₀
6		modxai.7	. . . . 5 ⊢ 𝐶 ∈ ℕ₀
7		expadd 10820	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶)))
8	4, 5, 6, 7	mp3an 1371	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶))
9	2, 8	eqtr3i 2252	. . 3 ⊢ (𝐴↑𝐸) = ((𝐴↑𝐵) · (𝐴↑𝐶))
10	9	oveq1i 6020	. 2 ⊢ ((𝐴↑𝐸) mod 𝑁) = (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁)
11		nnexpcl 10791	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ₀) → (𝐴↑𝐵) ∈ ℕ)
12	3, 5, 11	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐵) ∈ ℕ
13	12	nnzi 9483	. . . . . . 7 ⊢ (𝐴↑𝐵) ∈ ℤ
14	13	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐵) ∈ ℤ)
15		modxai.5	. . . . . . . 8 ⊢ 𝐾 ∈ ℕ₀
16	15	nn0zi 9484	. . . . . . 7 ⊢ 𝐾 ∈ ℤ
17	16	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐾 ∈ ℤ)
18		nnexpcl 10791	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑𝐶) ∈ ℕ)
19	3, 6, 18	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐶) ∈ ℕ
20	19	nnzi 9483	. . . . . . 7 ⊢ (𝐴↑𝐶) ∈ ℤ
21	20	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐶) ∈ ℤ)
22		modxai.8	. . . . . . . 8 ⊢ 𝐿 ∈ ℕ₀
23	22	nn0zi 9484	. . . . . . 7 ⊢ 𝐿 ∈ ℤ
24	23	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐿 ∈ ℤ)
25		modxai.1	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
26		nnq 9845	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
27	25, 26	mp1i 10	. . . . . 6 ⊢ (⊤ → 𝑁 ∈ ℚ)
28		nnrp 9876	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
29	25, 28	mp1i 10	. . . . . . 7 ⊢ (⊤ → 𝑁 ∈ ℝ⁺)
30	29	rpgt0d 9912	. . . . . 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 10617	. . . . 5 ⊢ (⊤ → (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁))
36	35	mptru 1404	. . . 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 9467	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ)
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ 𝐷 ∈ ℂ
41	25	nncni 9136	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
42	40, 41	mulcli 8167	. . . . . . 7 ⊢ (𝐷 · 𝑁) ∈ ℂ
43		modxai.6	. . . . . . . 8 ⊢ 𝑀 ∈ ℕ₀
44	43	nn0cni 9397	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
45	42, 44	addcomi 8306	. . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝑀 + (𝐷 · 𝑁))
46	37, 45	eqtr3i 2252	. . . . 5 ⊢ (𝐾 · 𝐿) = (𝑀 + (𝐷 · 𝑁))
47	46	oveq1i 6020	. . . 4 ⊢ ((𝐾 · 𝐿) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
48	36, 47	eqtri 2250	. . 3 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
49		nn0z 9482	. . . . 5 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
50		zq 9838	. . . . 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 1404	. . . 4 ⊢ 0 < 𝑁
54		modqcyc 10598	. . . 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 2250	. 2 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = (𝑀 mod 𝑁)
57	10, 56	eqtri 2250	1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Colors of variables: wff set class

Syntax hints: = wceq 1395 ⊤wtru 1396 ∈ wcel 2200 class class class wbr 4083 (class class class)co 6010 ℂcc 8013 0cc0 8015 + caddc 8018 · cmul 8020 < clt 8197 ℕcn 9126 ℕ₀cn0 9385 ℤcz 9462 ℚcq 9831 ℝ⁺crp 9866 mod cmo 10561 ↑cexp 10777

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-frec 6548 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-n0 9386 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-fl 10507 df-mod 10562 df-seqfrec 10687 df-exp 10778

This theorem is referenced by: mod2xi 12961 modxp1i 12962

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