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Mirrors > Home > MPE Home > Th. List > 2lgslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2lgs 25177: special case of 2lgs 25177 for 𝑃 = 2. (Contributed by AV, 20-Jun-2021.) |
Ref | Expression |
---|---|
2lgslem4 | ⊢ ((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lgs2 25175 | . . 3 ⊢ (2 /L 2) = 0 | |
2 | 1 | eqeq1i 2656 | . 2 ⊢ ((2 /L 2) = 1 ↔ 0 = 1) |
3 | 0ne1 11126 | . . . 4 ⊢ 0 ≠ 1 | |
4 | 3 | neii 2825 | . . 3 ⊢ ¬ 0 = 1 |
5 | 1ne2 11278 | . . . . 5 ⊢ 1 ≠ 2 | |
6 | 5 | nesymi 2880 | . . . 4 ⊢ ¬ 2 = 1 |
7 | 2re 11128 | . . . . . 6 ⊢ 2 ∈ ℝ | |
8 | 2lt7 11251 | . . . . . 6 ⊢ 2 < 7 | |
9 | 7, 8 | ltneii 10188 | . . . . 5 ⊢ 2 ≠ 7 |
10 | 9 | neii 2825 | . . . 4 ⊢ ¬ 2 = 7 |
11 | 6, 10 | pm3.2ni 917 | . . 3 ⊢ ¬ (2 = 1 ∨ 2 = 7) |
12 | 4, 11 | 2false 364 | . 2 ⊢ (0 = 1 ↔ (2 = 1 ∨ 2 = 7)) |
13 | 8nn 11229 | . . . . . 6 ⊢ 8 ∈ ℕ | |
14 | nnrp 11880 | . . . . . 6 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 8 ∈ ℝ+ |
16 | 0le2 11149 | . . . . 5 ⊢ 0 ≤ 2 | |
17 | 2lt8 11258 | . . . . 5 ⊢ 2 < 8 | |
18 | modid 12735 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 8 ∈ ℝ+) ∧ (0 ≤ 2 ∧ 2 < 8)) → (2 mod 8) = 2) | |
19 | 7, 15, 16, 17, 18 | mp4an 709 | . . . 4 ⊢ (2 mod 8) = 2 |
20 | 19 | eleq1i 2721 | . . 3 ⊢ ((2 mod 8) ∈ {1, 7} ↔ 2 ∈ {1, 7}) |
21 | 2ex 11130 | . . . 4 ⊢ 2 ∈ V | |
22 | 21 | elpr 4231 | . . 3 ⊢ (2 ∈ {1, 7} ↔ (2 = 1 ∨ 2 = 7)) |
23 | 20, 22 | bitr2i 265 | . 2 ⊢ ((2 = 1 ∨ 2 = 7) ↔ (2 mod 8) ∈ {1, 7}) |
24 | 2, 12, 23 | 3bitri 286 | 1 ⊢ ((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 382 = wceq 1523 ∈ wcel 2030 {cpr 4212 class class class wbr 4685 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 < clt 10112 ≤ cle 10113 ℕcn 11058 2c2 11108 7c7 11113 8c8 11114 ℝ+crp 11870 mod cmo 12708 /L clgs 25064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-dvds 15028 df-gcd 15264 df-prm 15433 df-phi 15518 df-pc 15589 df-lgs 25065 |
This theorem is referenced by: 2lgs 25177 |
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