4sqlem6 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 4sqlem6		Structured version Visualization version GIF version

Theorem 4sqlem6 16990

Description: Lemma for 4sq 17011. (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 11293	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
2		4sqlem5.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ)
3	2	zred 12747	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		4sqlem5.3	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
5	4	nnred 12308	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
6	5	rehalfcld 12540	. . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ)
7	3, 6	readdcld 11319	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ)
8	4	nnrpd 13097	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
9	7, 8	modcld 13926	. . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
10		modge0 13930	. . . . 5 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀))
11	7, 8, 10	syl2anc 583	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀))
12	1, 9, 6, 11	lesub1dd 11906	. . 3 ⊢ (𝜑 → (0 − (𝑀 / 2)) ≤ (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)))
13		df-neg 11523	. . 3 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
14		4sqlem5.4	. . 3 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
15	12, 13, 14	3brtr4g 5200	. 2 ⊢ (𝜑 → -(𝑀 / 2) ≤ 𝐵)
16		modlt 13931	. . . . . 6 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀)
17	7, 8, 16	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀)
18	4	nncnd 12309	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
19	18	2halvesd 12539	. . . . 5 ⊢ (𝜑 → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
20	17, 19	breqtrrd 5194	. . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2)))
21	9, 6, 6	ltsubaddd 11886	. . . 4 ⊢ (𝜑 → ((((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2))))
22	20, 21	mpbird 257	. . 3 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2))
23	14, 22	eqbrtrid 5201	. 2 ⊢ (𝜑 → 𝐵 < (𝑀 / 2))
24	15, 23	jca 511	1 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 class class class wbr 5166 (class class class)co 7448 ℝcr 11183 0cc0 11184 + caddc 11187 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 ℤcz 12639 ℝ⁺crp 13057 mod cmo 13920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843 df-mod 13921

This theorem is referenced by: 4sqlem7 16991 4sqlem10 16994

W3C validator