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Mirrors > Home > MPE Home > Th. List > 4sqlem5 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 15668. (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 11483 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 4sqlem5.4 | . . . . 5 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
4 | 1 | zred 11482 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4sqlem5.3 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
6 | 5 | nnred 11035 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | 6 | rehalfcld 11279 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
8 | 4, 7 | readdcld 10069 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
9 | 5 | nnrpd 11870 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
10 | 8, 9 | modcld 12674 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
11 | 10 | recnd 10068 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ) |
12 | 7 | recnd 10068 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℂ) |
13 | 11, 12 | subcld 10392 | . . . . 5 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) ∈ ℂ) |
14 | 3, 13 | syl5eqel 2705 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 2, 14 | nncand 10397 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
16 | 2, 14 | subcld 10392 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
17 | 6 | recnd 10068 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | nnne0d 11065 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) |
19 | 16, 17, 18 | divcan1d 10802 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) = (𝐴 − 𝐵)) |
20 | 3 | oveq2i 6661 | . . . . . . . . 9 ⊢ (𝐴 − 𝐵) = (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) |
21 | 2, 11, 12 | subsub3d 10422 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
22 | 20, 21 | syl5eq 2668 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝐵) = ((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀))) |
23 | 22 | oveq1d 6665 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) = (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀)) |
24 | moddifz 12682 | . . . . . . . 8 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) | |
25 | 8, 9, 24 | syl2anc 693 | . . . . . . 7 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) − ((𝐴 + (𝑀 / 2)) mod 𝑀)) / 𝑀) ∈ ℤ) |
26 | 23, 25 | eqeltrd 2701 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
27 | 5 | nnzd 11481 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
28 | 26, 27 | zmulcld 11488 | . . . . 5 ⊢ (𝜑 → (((𝐴 − 𝐵) / 𝑀) · 𝑀) ∈ ℤ) |
29 | 19, 28 | eqeltrrd 2702 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
30 | 1, 29 | zsubcld 11487 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) ∈ ℤ) |
31 | 15, 30 | eqeltrrd 2702 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
32 | 31, 26 | jca 554 | 1 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 ℝcr 9935 + caddc 9939 · cmul 9941 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℤcz 11377 ℝ+crp 11832 mod cmo 12668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 df-mod 12669 |
This theorem is referenced by: 4sqlem7 15648 4sqlem8 15649 4sqlem9 15650 4sqlem10 15651 4sqlem14 15662 4sqlem15 15663 4sqlem16 15664 4sqlem17 15665 2sqlem8a 25150 2sqlem8 25151 |
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