lgsdir2lem2 - Metamath Proof Explorer

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Theorem lgsdir2lem2 25904

Description: Lemma for lgsdir2 25908. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Hypotheses**
Ref	Expression
lgsdir2lem2.1	⊢ (𝐾 ∈ ℤ ∧ 2 ∥ (𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆)))
lgsdir2lem2.2	⊢ 𝑀 = (𝐾 + 1)
lgsdir2lem2.3	⊢ 𝑁 = (𝑀 + 1)
lgsdir2lem2.4	⊢ 𝑁 ∈ 𝑆

**Assertion**
Ref	Expression
lgsdir2lem2	⊢ (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆)))

**Proof of Theorem lgsdir2lem2**
Step	Hyp	Ref	Expression
1		lgsdir2lem2.3	. . 3 ⊢ 𝑁 = (𝑀 + 1)
2		lgsdir2lem2.2	. . . . 5 ⊢ 𝑀 = (𝐾 + 1)
3		lgsdir2lem2.1	. . . . . . 7 ⊢ (𝐾 ∈ ℤ ∧ 2 ∥ (𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆)))
4	3	simp1i 1135	. . . . . 6 ⊢ 𝐾 ∈ ℤ
5		peano2z 12026	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 + 1) ∈ ℤ)
6	4, 5	ax-mp 5	. . . . 5 ⊢ (𝐾 + 1) ∈ ℤ
7	2, 6	eqeltri 2911	. . . 4 ⊢ 𝑀 ∈ ℤ
8		peano2z 12026	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
9	7, 8	ax-mp 5	. . 3 ⊢ (𝑀 + 1) ∈ ℤ
10	1, 9	eqeltri 2911	. 2 ⊢ 𝑁 ∈ ℤ
11	3	simp2i 1136	. . . 4 ⊢ 2 ∥ (𝐾 + 1)
12		2z 12017	. . . . 5 ⊢ 2 ∈ ℤ
13		dvdsadd 15654	. . . . 5 ⊢ ((2 ∈ ℤ ∧ (𝐾 + 1) ∈ ℤ) → (2 ∥ (𝐾 + 1) ↔ 2 ∥ (2 + (𝐾 + 1))))
14	12, 6, 13	mp2an 690	. . . 4 ⊢ (2 ∥ (𝐾 + 1) ↔ 2 ∥ (2 + (𝐾 + 1)))
15	11, 14	mpbi 232	. . 3 ⊢ 2 ∥ (2 + (𝐾 + 1))
16		zcn 11989	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
17	4, 16	ax-mp 5	. . . . . . . . . 10 ⊢ 𝐾 ∈ ℂ
18		ax-1cn 10597	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
19	17, 18	addcomi 10833	. . . . . . . . 9 ⊢ (𝐾 + 1) = (1 + 𝐾)
20	2, 19	eqtri 2846	. . . . . . . 8 ⊢ 𝑀 = (1 + 𝐾)
21	20	oveq1i 7168	. . . . . . 7 ⊢ (𝑀 + 1) = ((1 + 𝐾) + 1)
22	1, 21	eqtri 2846	. . . . . 6 ⊢ 𝑁 = ((1 + 𝐾) + 1)
23		df-2 11703	. . . . . . . 8 ⊢ 2 = (1 + 1)
24	23	oveq1i 7168	. . . . . . 7 ⊢ (2 + 𝐾) = ((1 + 1) + 𝐾)
25	18, 17, 18	add32i 10865	. . . . . . 7 ⊢ ((1 + 𝐾) + 1) = ((1 + 1) + 𝐾)
26	24, 25	eqtr4i 2849	. . . . . 6 ⊢ (2 + 𝐾) = ((1 + 𝐾) + 1)
27	22, 26	eqtr4i 2849	. . . . 5 ⊢ 𝑁 = (2 + 𝐾)
28	27	oveq1i 7168	. . . 4 ⊢ (𝑁 + 1) = ((2 + 𝐾) + 1)
29		2cn 11715	. . . . 5 ⊢ 2 ∈ ℂ
30	29, 17, 18	addassi 10653	. . . 4 ⊢ ((2 + 𝐾) + 1) = (2 + (𝐾 + 1))
31	28, 30	eqtri 2846	. . 3 ⊢ (𝑁 + 1) = (2 + (𝐾 + 1))
32	15, 31	breqtrri 5095	. 2 ⊢ 2 ∥ (𝑁 + 1)
33		elfzuz2 12915	. . . . 5 ⊢ ((𝐴 mod 8) ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘0))
34		fzm1 12990	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘0) → ((𝐴 mod 8) ∈ (0...𝑁) ↔ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁)))
35	33, 34	syl 17	. . . 4 ⊢ ((𝐴 mod 8) ∈ (0...𝑁) → ((𝐴 mod 8) ∈ (0...𝑁) ↔ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁)))
36	35	ibi 269	. . 3 ⊢ ((𝐴 mod 8) ∈ (0...𝑁) → ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁))
37		elfzuz2 12915	. . . . . . . 8 ⊢ ((𝐴 mod 8) ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘0))
38		fzm1 12990	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → ((𝐴 mod 8) ∈ (0...𝑀) ↔ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)))
39	37, 38	syl 17	. . . . . . 7 ⊢ ((𝐴 mod 8) ∈ (0...𝑀) → ((𝐴 mod 8) ∈ (0...𝑀) ↔ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)))
40	39	ibi 269	. . . . . 6 ⊢ ((𝐴 mod 8) ∈ (0...𝑀) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀))
41		zcn 11989	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
42	7, 41	ax-mp 5	. . . . . . . 8 ⊢ 𝑀 ∈ ℂ
43	42, 18, 1	mvrraddi 10905	. . . . . . 7 ⊢ (𝑁 − 1) = 𝑀
44	43	oveq2i 7169	. . . . . 6 ⊢ (0...(𝑁 − 1)) = (0...𝑀)
45	40, 44	eleq2s 2933	. . . . 5 ⊢ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀))
46	17, 18, 2	mvrraddi 10905	. . . . . . . . 9 ⊢ (𝑀 − 1) = 𝐾
47	46	oveq2i 7169	. . . . . . . 8 ⊢ (0...(𝑀 − 1)) = (0...𝐾)
48	47	eleq2i 2906	. . . . . . 7 ⊢ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ↔ (𝐴 mod 8) ∈ (0...𝐾))
49	3	simp3i 1137	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆))
50	48, 49	syl5bi 244	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) → (𝐴 mod 8) ∈ 𝑆))
51		2nn 11713	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
52		8nn 11735	. . . . . . . . . . 11 ⊢ 8 ∈ ℕ
53		4z 12019	. . . . . . . . . . . . . 14 ⊢ 4 ∈ ℤ
54		dvdsmul2 15634	. . . . . . . . . . . . . 14 ⊢ ((4 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (4 · 2))
55	53, 12, 54	mp2an 690	. . . . . . . . . . . . 13 ⊢ 2 ∥ (4 · 2)
56		4t2e8 11808	. . . . . . . . . . . . 13 ⊢ (4 · 2) = 8
57	55, 56	breqtri 5093	. . . . . . . . . . . 12 ⊢ 2 ∥ 8
58		dvdsmod 15680	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℕ ∧ 8 ∈ ℕ ∧ 𝐴 ∈ ℤ) ∧ 2 ∥ 8) → (2 ∥ (𝐴 mod 8) ↔ 2 ∥ 𝐴))
59	57, 58	mpan2 689	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 8 ∈ ℕ ∧ 𝐴 ∈ ℤ) → (2 ∥ (𝐴 mod 8) ↔ 2 ∥ 𝐴))
60	51, 52, 59	mp3an12 1447	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → (2 ∥ (𝐴 mod 8) ↔ 2 ∥ 𝐴))
61	60	notbid 320	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (¬ 2 ∥ (𝐴 mod 8) ↔ ¬ 2 ∥ 𝐴))
62	61	biimpar 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ¬ 2 ∥ (𝐴 mod 8))
63	11, 2	breqtrri 5095	. . . . . . . . 9 ⊢ 2 ∥ 𝑀
64		id 22	. . . . . . . . 9 ⊢ ((𝐴 mod 8) = 𝑀 → (𝐴 mod 8) = 𝑀)
65	63, 64	breqtrrid 5106	. . . . . . . 8 ⊢ ((𝐴 mod 8) = 𝑀 → 2 ∥ (𝐴 mod 8))
66	62, 65	nsyl 142	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ¬ (𝐴 mod 8) = 𝑀)
67	66	pm2.21d 121	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) = 𝑀 → (𝐴 mod 8) ∈ 𝑆))
68	50, 67	jaod 855	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀) → (𝐴 mod 8) ∈ 𝑆))
69	45, 68	syl5 34	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) → (𝐴 mod 8) ∈ 𝑆))
70		lgsdir2lem2.4	. . . . . 6 ⊢ 𝑁 ∈ 𝑆
71		eleq1 2902	. . . . . 6 ⊢ ((𝐴 mod 8) = 𝑁 → ((𝐴 mod 8) ∈ 𝑆 ↔ 𝑁 ∈ 𝑆))
72	70, 71	mpbiri 260	. . . . 5 ⊢ ((𝐴 mod 8) = 𝑁 → (𝐴 mod 8) ∈ 𝑆)
73	72	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) = 𝑁 → (𝐴 mod 8) ∈ 𝑆))
74	69, 73	jaod 855	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁) → (𝐴 mod 8) ∈ 𝑆))
75	36, 74	syl5 34	. 2 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆))
76	10, 32, 75	3pm3.2i 1335	1 ⊢ (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 ℕcn 11640 2c2 11695 4c4 11697 8c8 11701 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 mod cmo 13240 ∥ cdvds 15609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fl 13165 df-mod 13241 df-dvds 15610

This theorem is referenced by: lgsdir2lem3 25905

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