4sqlem6 - Metamath Proof Explorer

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Theorem 4sqlem6 16644

Description: Lemma for 4sq 16665. (Contributed by Mario Carneiro, 15-Jul-2014.)

**Hypotheses**
Ref	Expression
4sqlem5.2	⊢ (𝜑 → 𝐴 ∈ ℤ)
4sqlem5.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
4sqlem5.4	⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))

**Assertion**
Ref	Expression
4sqlem6	⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))

**Proof of Theorem 4sqlem6**
Step	Hyp	Ref	Expression
1		0red 10978	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
2		4sqlem5.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ)
3	2	zred 12426	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		4sqlem5.3	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
5	4	nnred 11988	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
6	5	rehalfcld 12220	. . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ)
7	3, 6	readdcld 11004	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ)
8	4	nnrpd 12770	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
9	7, 8	modcld 13595	. . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
10		modge0 13599	. . . . 5 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀))
11	7, 8, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀))
12	1, 9, 6, 11	lesub1dd 11591	. . 3 ⊢ (𝜑 → (0 − (𝑀 / 2)) ≤ (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)))
13		df-neg 11208	. . 3 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
14		4sqlem5.4	. . 3 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
15	12, 13, 14	3brtr4g 5108	. 2 ⊢ (𝜑 → -(𝑀 / 2) ≤ 𝐵)
16		modlt 13600	. . . . . 6 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀)
17	7, 8, 16	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀)
18	4	nncnd 11989	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
19	18	2halvesd 12219	. . . . 5 ⊢ (𝜑 → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
20	17, 19	breqtrrd 5102	. . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2)))
21	9, 6, 6	ltsubaddd 11571	. . . 4 ⊢ (𝜑 → ((((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2))))
22	20, 21	mpbird 256	. . 3 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2))
23	14, 22	eqbrtrid 5109	. 2 ⊢ (𝜑 → 𝐵 < (𝑀 / 2))
24	15, 23	jca 512	1 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 0cc0 10871 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 2c2 12028 ℤcz 12319 ℝ⁺crp 12730 mod cmo 13589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fl 13512 df-mod 13590

This theorem is referenced by: 4sqlem7 16645 4sqlem10 16648

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