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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0f | Structured version Visualization version GIF version |
Description: Auxiliary lemma 6 for gausslemma2d 25632. (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 4626 | . . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) | |
3 | prm23ge5 15981 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) | |
4 | eqneqall 2995 | . . . . . . 7 ⊢ (𝑃 = 2 → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) | |
5 | orc 862 | . . . . . . . 8 ⊢ (𝑃 = 3 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) | |
6 | 5 | a1d 25 | . . . . . . 7 ⊢ (𝑃 = 3 → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
7 | olc 863 | . . . . . . . 8 ⊢ (𝑃 ∈ (ℤ≥‘5) → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) | |
8 | 7 | a1d 25 | . . . . . . 7 ⊢ (𝑃 ∈ (ℤ≥‘5) → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
9 | 4, 6, 8 | 3jaoi 1420 | . . . . . 6 ⊢ ((𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)) → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
10 | 3, 9 | syl 17 | . . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 ≠ 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
11 | 10 | imp 407 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) |
12 | 2, 11 | sylbi 218 | . . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) |
13 | fldiv4p1lem1div2 13055 | . . 3 ⊢ ((𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)) → ((⌊‘(𝑃 / 4)) + 1) ≤ ((𝑃 − 1) / 2)) | |
14 | 1, 12, 13 | 3syl 18 | . 2 ⊢ (𝜑 → ((⌊‘(𝑃 / 4)) + 1) ≤ ((𝑃 − 1) / 2)) |
15 | gausslemma2dlem0.m | . . 3 ⊢ 𝑀 = (⌊‘(𝑃 / 4)) | |
16 | 15 | oveq1i 7026 | . 2 ⊢ (𝑀 + 1) = ((⌊‘(𝑃 / 4)) + 1) |
17 | gausslemma2dlem0.h | . 2 ⊢ 𝐻 = ((𝑃 − 1) / 2) | |
18 | 14, 16, 17 | 3brtr4g 4996 | 1 ⊢ (𝜑 → (𝑀 + 1) ≤ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 842 ∨ w3o 1079 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∖ cdif 3856 {csn 4472 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 1c1 10384 + caddc 10386 ≤ cle 10522 − cmin 10717 / cdiv 11145 2c2 11540 3c3 11541 4c4 11542 5c5 11543 ℤ≥cuz 12093 ⌊cfl 13010 ℙcprime 15844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-fz 12743 df-fl 13012 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-dvds 15441 df-prm 15845 |
This theorem is referenced by: gausslemma2dlem5 25629 |
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