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Mirrors > Home > ILE Home > Th. List > gausslemma2dlem1cl | GIF version |
Description: Lemma for gausslemma2dlem1 15119. Closure of the body of the definition of 𝑅. (Contributed by Jim Kingdon, 10-Aug-2025.) |
Ref | Expression |
---|---|
gausslemma2d.p | ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) |
gausslemma2d.h | ⊢ 𝐻 = ((𝑃 − 1) / 2) |
gausslemma2d.r | ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) |
gausslemma2dlem1cl.a | ⊢ (𝜑 → 𝐴 ∈ (1...𝐻)) |
Ref | Expression |
---|---|
gausslemma2dlem1cl | ⊢ (𝜑 → if((𝐴 · 2) < (𝑃 / 2), (𝐴 · 2), (𝑃 − (𝐴 · 2))) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gausslemma2dlem1cl.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (1...𝐻)) | |
2 | 1 | elfzelzd 10082 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
3 | 2z 9335 | . . . 4 ⊢ 2 ∈ ℤ | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝜑 → 2 ∈ ℤ) |
5 | 2, 4 | zmulcld 9435 | . 2 ⊢ (𝜑 → (𝐴 · 2) ∈ ℤ) |
6 | gausslemma2d.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
7 | eldifi 3281 | . . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
8 | prmz 12236 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
9 | 6, 7, 8 | 3syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
10 | 9, 5 | zsubcld 9434 | . 2 ⊢ (𝜑 → (𝑃 − (𝐴 · 2)) ∈ ℤ) |
11 | zq 9681 | . . . 4 ⊢ ((𝐴 · 2) ∈ ℤ → (𝐴 · 2) ∈ ℚ) | |
12 | 5, 11 | syl 14 | . . 3 ⊢ (𝜑 → (𝐴 · 2) ∈ ℚ) |
13 | 2nn 9133 | . . . . 5 ⊢ 2 ∈ ℕ | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝜑 → 2 ∈ ℕ) |
15 | znq 9679 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑃 / 2) ∈ ℚ) | |
16 | 9, 14, 15 | syl2anc 411 | . . 3 ⊢ (𝜑 → (𝑃 / 2) ∈ ℚ) |
17 | qdclt 10305 | . . 3 ⊢ (((𝐴 · 2) ∈ ℚ ∧ (𝑃 / 2) ∈ ℚ) → DECID (𝐴 · 2) < (𝑃 / 2)) | |
18 | 12, 16, 17 | syl2anc 411 | . 2 ⊢ (𝜑 → DECID (𝐴 · 2) < (𝑃 / 2)) |
19 | 5, 10, 18 | ifcldcd 3593 | 1 ⊢ (𝜑 → if((𝐴 · 2) < (𝑃 / 2), (𝐴 · 2), (𝑃 − (𝐴 · 2))) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 835 = wceq 1364 ∈ wcel 2164 ∖ cdif 3150 ifcif 3557 {csn 3618 class class class wbr 4029 ↦ cmpt 4090 (class class class)co 5910 1c1 7863 · cmul 7867 < clt 8044 − cmin 8180 / cdiv 8681 ℕcn 8972 2c2 9023 ℤcz 9307 ℚcq 9674 ...cfz 10064 ℙcprime 12232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-pre-mulext 7980 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-po 4325 df-iso 4326 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-ap 8591 df-div 8682 df-inn 8973 df-2 9031 df-n0 9231 df-z 9308 df-uz 9583 df-q 9675 df-rp 9710 df-fz 10065 df-prm 12233 |
This theorem is referenced by: gausslemma2dlem1f1o 15118 |
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