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Mirrors > Home > MPE Home > Th. List > 4sqlem5 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16294. (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 12082 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 4sqlem5.4 | . . . . 5 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
4 | 1 | zred 12081 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4sqlem5.3 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
6 | 5 | nnred 11647 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | 6 | rehalfcld 11878 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
8 | 4, 7 | readdcld 10664 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
9 | 5 | nnrpd 12423 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
10 | 8, 9 | modcld 13237 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
11 | 10 | recnd 10663 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ) |
12 | 7 | recnd 10663 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℂ) |
13 | 11, 12 | subcld 10991 | . . . . 5 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) ∈ ℂ) |
14 | 3, 13 | eqeltrid 2917 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 2, 14 | nncand 10996 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
16 | 2, 14 | subcld 10991 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
17 | 6 | recnd 10663 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | nnne0d 11681 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) |
19 | 16, 17, 18 | divcan1d 11411 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) = (𝐴 − 𝐵)) |
20 | 3 | oveq2i 7161 | . . . . . . . . 9 ⊢ (𝐴 − 𝐵) = (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) |
21 | 2, 11, 12 | subsub3d 11021 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
22 | 20, 21 | syl5eq 2868 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝐵) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
23 | 22 | oveq1d 7165 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) = (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀)) |
24 | moddifz 13245 | . . . . . . . 8 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) | |
25 | 8, 9, 24 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) |
26 | 23, 25 | eqeltrd 2913 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
27 | 5 | nnzd 12080 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
28 | 26, 27 | zmulcld 12087 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) ∈ ℤ) |
29 | 19, 28 | eqeltrrd 2914 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
30 | 1, 29 | zsubcld 12086 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) ∈ ℤ) |
31 | 15, 30 | eqeltrrd 2914 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
32 | 31, 26 | jca 514 | 1 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 ℝcr 10530 + caddc 10534 · cmul 10536 − cmin 10864 / cdiv 11291 ℕcn 11632 2c2 11686 ℤcz 11975 ℝ+crp 12383 mod cmo 13231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13156 df-mod 13232 |
This theorem is referenced by: 4sqlem7 16274 4sqlem8 16275 4sqlem9 16276 4sqlem10 16277 4sqlem14 16288 4sqlem15 16289 4sqlem16 16290 4sqlem17 16291 2sqlem8a 25995 2sqlem8 25996 |
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