modxai - Intuitionistic Logic Explorer

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Theorem modxai 12610

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 5936	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = (𝐴↑𝐸)
3		modxai.2	. . . . . 6 ⊢ 𝐴 ∈ ℕ
4	3	nncni 9017	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		modxai.3	. . . . 5 ⊢ 𝐵 ∈ ℕ₀
6		modxai.7	. . . . 5 ⊢ 𝐶 ∈ ℕ₀
7		expadd 10690	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶)))
8	4, 5, 6, 7	mp3an 1348	. . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶))
9	2, 8	eqtr3i 2219	. . 3 ⊢ (𝐴↑𝐸) = ((𝐴↑𝐵) · (𝐴↑𝐶))
10	9	oveq1i 5935	. 2 ⊢ ((𝐴↑𝐸) mod 𝑁) = (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁)
11		nnexpcl 10661	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ₀) → (𝐴↑𝐵) ∈ ℕ)
12	3, 5, 11	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐵) ∈ ℕ
13	12	nnzi 9364	. . . . . . 7 ⊢ (𝐴↑𝐵) ∈ ℤ
14	13	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐵) ∈ ℤ)
15		modxai.5	. . . . . . . 8 ⊢ 𝐾 ∈ ℕ₀
16	15	nn0zi 9365	. . . . . . 7 ⊢ 𝐾 ∈ ℤ
17	16	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐾 ∈ ℤ)
18		nnexpcl 10661	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ₀) → (𝐴↑𝐶) ∈ ℕ)
19	3, 6, 18	mp2an 426	. . . . . . . 8 ⊢ (𝐴↑𝐶) ∈ ℕ
20	19	nnzi 9364	. . . . . . 7 ⊢ (𝐴↑𝐶) ∈ ℤ
21	20	a1i 9	. . . . . 6 ⊢ (⊤ → (𝐴↑𝐶) ∈ ℤ)
22		modxai.8	. . . . . . . 8 ⊢ 𝐿 ∈ ℕ₀
23	22	nn0zi 9365	. . . . . . 7 ⊢ 𝐿 ∈ ℤ
24	23	a1i 9	. . . . . 6 ⊢ (⊤ → 𝐿 ∈ ℤ)
25		modxai.1	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
26		nnq 9724	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
27	25, 26	mp1i 10	. . . . . 6 ⊢ (⊤ → 𝑁 ∈ ℚ)
28		nnrp 9755	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
29	25, 28	mp1i 10	. . . . . . 7 ⊢ (⊤ → 𝑁 ∈ ℝ⁺)
30	29	rpgt0d 9791	. . . . . 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 10487	. . . . 5 ⊢ (⊤ → (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁))
36	35	mptru 1373	. . . 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 9348	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ)
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ 𝐷 ∈ ℂ
41	25	nncni 9017	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
42	40, 41	mulcli 8048	. . . . . . 7 ⊢ (𝐷 · 𝑁) ∈ ℂ
43		modxai.6	. . . . . . . 8 ⊢ 𝑀 ∈ ℕ₀
44	43	nn0cni 9278	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
45	42, 44	addcomi 8187	. . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝑀 + (𝐷 · 𝑁))
46	37, 45	eqtr3i 2219	. . . . 5 ⊢ (𝐾 · 𝐿) = (𝑀 + (𝐷 · 𝑁))
47	46	oveq1i 5935	. . . 4 ⊢ ((𝐾 · 𝐿) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
48	36, 47	eqtri 2217	. . 3 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁)
49		nn0z 9363	. . . . 5 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
50		zq 9717	. . . . 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 1373	. . . 4 ⊢ 0 < 𝑁
54		modqcyc 10468	. . . 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 2217	. 2 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = (𝑀 mod 𝑁)
57	10, 56	eqtri 2217	1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Colors of variables: wff set class

Syntax hints: = wceq 1364 ⊤wtru 1365 ∈ wcel 2167 class class class wbr 4034 (class class class)co 5925 ℂcc 7894 0cc0 7896 + caddc 7899 · cmul 7901 < clt 8078 ℕcn 9007 ℕ₀cn0 9266 ℤcz 9343 ℚcq 9710 ℝ⁺crp 9745 mod cmo 10431 ↑cexp 10647

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-fl 10377 df-mod 10432 df-seqfrec 10557 df-exp 10648

This theorem is referenced by: mod2xi 12611 modxp1i 12612

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