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Mirrors > Home > MPE Home > Th. List > 4sqlem5 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 15890. (Contributed by Mario Carneiro, 15-Jul-2014.) |
Ref | Expression |
---|---|
4sqlem5.2 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
4sqlem5.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
4sqlem5.4 | ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) |
Ref | Expression |
---|---|
4sqlem5 | ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4sqlem5.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 1 | zcnd 11695 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 4sqlem5.4 | . . . . 5 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
4 | 1 | zred 11694 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4sqlem5.3 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
6 | 5 | nnred 11247 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | 6 | rehalfcld 11491 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
8 | 4, 7 | readdcld 10281 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
9 | 5 | nnrpd 12083 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
10 | 8, 9 | modcld 12888 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
11 | 10 | recnd 10280 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ) |
12 | 7 | recnd 10280 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℂ) |
13 | 11, 12 | subcld 10604 | . . . . 5 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) ∈ ℂ) |
14 | 3, 13 | syl5eqel 2843 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 2, 14 | nncand 10609 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
16 | 2, 14 | subcld 10604 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
17 | 6 | recnd 10280 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | nnne0d 11277 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) |
19 | 16, 17, 18 | divcan1d 11014 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) = (𝐴 − 𝐵)) |
20 | 3 | oveq2i 6825 | . . . . . . . . 9 ⊢ (𝐴 − 𝐵) = (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) |
21 | 2, 11, 12 | subsub3d 10634 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
22 | 20, 21 | syl5eq 2806 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝐵) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
23 | 22 | oveq1d 6829 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) = (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀)) |
24 | moddifz 12896 | . . . . . . . 8 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) | |
25 | 8, 9, 24 | syl2anc 696 | . . . . . . 7 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) |
26 | 23, 25 | eqeltrd 2839 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
27 | 5 | nnzd 11693 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
28 | 26, 27 | zmulcld 11700 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) ∈ ℤ) |
29 | 19, 28 | eqeltrrd 2840 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
30 | 1, 29 | zsubcld 11699 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) ∈ ℤ) |
31 | 15, 30 | eqeltrrd 2840 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
32 | 31, 26 | jca 555 | 1 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 (class class class)co 6814 ℂcc 10146 ℝcr 10147 + caddc 10151 · cmul 10153 − cmin 10478 / cdiv 10896 ℕcn 11232 2c2 11282 ℤcz 11589 ℝ+crp 12045 mod cmo 12882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-sup 8515 df-inf 8516 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-n0 11505 df-z 11590 df-uz 11900 df-rp 12046 df-fl 12807 df-mod 12883 |
This theorem is referenced by: 4sqlem7 15870 4sqlem8 15871 4sqlem9 15872 4sqlem10 15873 4sqlem14 15884 4sqlem15 15885 4sqlem16 15886 4sqlem17 15887 2sqlem8a 25370 2sqlem8 25371 |
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