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Mirrors > Home > MPE Home > Th. List > 4sqlem5 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16288. (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 12076 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 4sqlem5.4 | . . . . 5 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
4 | 1 | zred 12075 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4sqlem5.3 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
6 | 5 | nnred 11641 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | 6 | rehalfcld 11872 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
8 | 4, 7 | readdcld 10658 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
9 | 5 | nnrpd 12417 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
10 | 8, 9 | modcld 13231 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
11 | 10 | recnd 10657 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ) |
12 | 7 | recnd 10657 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℂ) |
13 | 11, 12 | subcld 10985 | . . . . 5 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) ∈ ℂ) |
14 | 3, 13 | eqeltrid 2914 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 2, 14 | nncand 10990 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
16 | 2, 14 | subcld 10985 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
17 | 6 | recnd 10657 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | nnne0d 11675 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) |
19 | 16, 17, 18 | divcan1d 11405 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) = (𝐴 − 𝐵)) |
20 | 3 | oveq2i 7156 | . . . . . . . . 9 ⊢ (𝐴 − 𝐵) = (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) |
21 | 2, 11, 12 | subsub3d 11015 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
22 | 20, 21 | syl5eq 2865 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝐵) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
23 | 22 | oveq1d 7160 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) = (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀)) |
24 | moddifz 13239 | . . . . . . . 8 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) | |
25 | 8, 9, 24 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) |
26 | 23, 25 | eqeltrd 2910 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
27 | 5 | nnzd 12074 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
28 | 26, 27 | zmulcld 12081 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) ∈ ℤ) |
29 | 19, 28 | eqeltrrd 2911 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
30 | 1, 29 | zsubcld 12080 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) ∈ ℤ) |
31 | 15, 30 | eqeltrrd 2911 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
32 | 31, 26 | jca 512 | 1 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 ℝcr 10524 + caddc 10528 · cmul 10530 − cmin 10858 / cdiv 11285 ℕcn 11626 2c2 11680 ℤcz 11969 ℝ+crp 12377 mod cmo 13225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-mod 13226 |
This theorem is referenced by: 4sqlem7 16268 4sqlem8 16269 4sqlem9 16270 4sqlem10 16271 4sqlem14 16282 4sqlem15 16283 4sqlem16 16284 4sqlem17 16285 2sqlem8a 25928 2sqlem8 25929 |
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