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Mirrors > Home > ILE Home > Th. List > gausslemma2dlem0f | GIF version |
Description: Auxiliary lemma 6 for gausslemma2d 15127. (Contributed by AV, 9-Jul-2021.) |
Ref | Expression |
---|---|
gausslemma2dlem0.p | ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) |
gausslemma2dlem0.m | ⊢ 𝑀 = (⌊‘(𝑃 / 4)) |
gausslemma2dlem0.h | ⊢ 𝐻 = ((𝑃 − 1) / 2) |
Ref | Expression |
---|---|
gausslemma2dlem0f | ⊢ (𝜑 → (𝑀 + 1) ≤ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gausslemma2dlem0.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
2 | eldifsn 3745 | . . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) | |
3 | prm23ge5 12389 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) | |
4 | eqneqall 2374 | . . . . . . 7 ⊢ (𝑃 = 2 → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) | |
5 | orc 713 | . . . . . . . 8 ⊢ (𝑃 = 3 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) | |
6 | 5 | a1d 22 | . . . . . . 7 ⊢ (𝑃 = 3 → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
7 | olc 712 | . . . . . . . 8 ⊢ (𝑃 ∈ (ℤ≥‘5) → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) | |
8 | 7 | a1d 22 | . . . . . . 7 ⊢ (𝑃 ∈ (ℤ≥‘5) → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
9 | 4, 6, 8 | 3jaoi 1314 | . . . . . 6 ⊢ ((𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)) → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
10 | 3, 9 | syl 14 | . . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
11 | 10 | imp 124 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) |
12 | 2, 11 | sylbi 121 | . . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) |
13 | fldiv4p1lem1div2 10364 | . . 3 ⊢ ((𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)) → ((⌊‘(𝑃 / 4)) + 1) ≤ ((𝑃 − 1) / 2)) | |
14 | 1, 12, 13 | 3syl 17 | . 2 ⊢ (𝜑 → ((⌊‘(𝑃 / 4)) + 1) ≤ ((𝑃 − 1) / 2)) |
15 | gausslemma2dlem0.m | . . 3 ⊢ 𝑀 = (⌊‘(𝑃 / 4)) | |
16 | 15 | oveq1i 5920 | . 2 ⊢ (𝑀 + 1) = ((⌊‘(𝑃 / 4)) + 1) |
17 | gausslemma2dlem0.h | . 2 ⊢ 𝐻 = ((𝑃 − 1) / 2) | |
18 | 14, 16, 17 | 3brtr4g 4063 | 1 ⊢ (𝜑 → (𝑀 + 1) ≤ 𝐻) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 709 ∨ w3o 979 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 ∖ cdif 3150 {csn 3618 class class class wbr 4029 ‘cfv 5246 (class class class)co 5910 1c1 7863 + caddc 7865 ≤ cle 8045 − cmin 8180 / cdiv 8681 2c2 9023 3c3 9024 4c4 9025 5c5 9026 ℤ≥cuz 9582 ⌊cfl 10327 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-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 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 ax-arch 7981 ax-caucvg 7982 |
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-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-tp 3626 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 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-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-recs 6349 df-frec 6435 df-1o 6460 df-2o 6461 df-er 6578 df-en 6786 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-3 9032 df-4 9033 df-5 9034 df-6 9035 df-n0 9231 df-z 9308 df-uz 9583 df-q 9675 df-rp 9710 df-fz 10065 df-fl 10329 df-seqfrec 10509 df-exp 10597 df-cj 10973 df-re 10974 df-im 10975 df-rsqrt 11129 df-abs 11130 df-dvds 11918 df-prm 12233 |
This theorem is referenced by: gausslemma2dlem5 15124 |
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