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Mirrors > Home > MPE Home > Th. List > 2lgslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2lgs 26596: special case of 2lgs 26596 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 26594 | . . 3 ⊢ (2 /L 2) = 0 | |
2 | 1 | eqeq1i 2741 | . 2 ⊢ ((2 /L 2) = 1 ↔ 0 = 1) |
3 | 0ne1 12086 | . . . 4 ⊢ 0 ≠ 1 | |
4 | 3 | neii 2943 | . . 3 ⊢ ¬ 0 = 1 |
5 | 1ne2 12223 | . . . . 5 ⊢ 1 ≠ 2 | |
6 | 5 | nesymi 2999 | . . . 4 ⊢ ¬ 2 = 1 |
7 | 2re 12089 | . . . . . 6 ⊢ 2 ∈ ℝ | |
8 | 2lt7 12205 | . . . . . 6 ⊢ 2 < 7 | |
9 | 7, 8 | ltneii 11130 | . . . . 5 ⊢ 2 ≠ 7 |
10 | 9 | neii 2943 | . . . 4 ⊢ ¬ 2 = 7 |
11 | 6, 10 | pm3.2ni 879 | . . 3 ⊢ ¬ (2 = 1 ∨ 2 = 7) |
12 | 4, 11 | 2false 377 | . 2 ⊢ (0 = 1 ↔ (2 = 1 ∨ 2 = 7)) |
13 | 8nn 12110 | . . . . . 6 ⊢ 8 ∈ ℕ | |
14 | nnrp 12783 | . . . . . 6 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 8 ∈ ℝ+ |
16 | 0le2 12117 | . . . . 5 ⊢ 0 ≤ 2 | |
17 | 2lt8 12212 | . . . . 5 ⊢ 2 < 8 | |
18 | modid 13658 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 8 ∈ ℝ+) ∧ (0 ≤ 2 ∧ 2 < 8)) → (2 mod 8) = 2) | |
19 | 7, 15, 16, 17, 18 | mp4an 691 | . . . 4 ⊢ (2 mod 8) = 2 |
20 | 19 | eleq1i 2827 | . . 3 ⊢ ((2 mod 8) ∈ {1, 7} ↔ 2 ∈ {1, 7}) |
21 | 2ex 12092 | . . . 4 ⊢ 2 ∈ V | |
22 | 21 | elpr 4588 | . . 3 ⊢ (2 ∈ {1, 7} ↔ (2 = 1 ∨ 2 = 7)) |
23 | 20, 22 | bitr2i 277 | . 2 ⊢ ((2 = 1 ∨ 2 = 7) ↔ (2 mod 8) ∈ {1, 7}) |
24 | 2, 12, 23 | 3bitri 298 | 1 ⊢ ((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 845 = wceq 1539 ∈ wcel 2104 {cpr 4567 class class class wbr 5081 (class class class)co 7303 ℝcr 10912 0cc0 10913 1c1 10914 < clt 11051 ≤ cle 11052 ℕcn 12015 2c2 12070 7c7 12075 8c8 12076 ℝ+crp 12772 mod cmo 13631 /L clgs 26483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-2o 8325 df-oadd 8328 df-er 8525 df-map 8644 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-sup 9241 df-inf 9242 df-dju 9699 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-4 12080 df-5 12081 df-6 12082 df-7 12083 df-8 12084 df-n0 12276 df-xnn0 12348 df-z 12362 df-uz 12625 df-q 12731 df-rp 12773 df-fz 13282 df-fzo 13425 df-fl 13554 df-mod 13632 df-seq 13764 df-exp 13825 df-hash 14087 df-cj 14851 df-re 14852 df-im 14853 df-sqrt 14987 df-abs 14988 df-dvds 16005 df-gcd 16243 df-prm 16418 df-phi 16508 df-pc 16579 df-lgs 26484 |
This theorem is referenced by: 2lgs 26596 |
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