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Mirrors > Home > MPE Home > Th. List > 2lgslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2lgs 25983: special case of 2lgs 25983 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 25981 | . . 3 ⊢ (2 /L 2) = 0 | |
2 | 1 | eqeq1i 2826 | . 2 ⊢ ((2 /L 2) = 1 ↔ 0 = 1) |
3 | 0ne1 11709 | . . . 4 ⊢ 0 ≠ 1 | |
4 | 3 | neii 3018 | . . 3 ⊢ ¬ 0 = 1 |
5 | 1ne2 11846 | . . . . 5 ⊢ 1 ≠ 2 | |
6 | 5 | nesymi 3073 | . . . 4 ⊢ ¬ 2 = 1 |
7 | 2re 11712 | . . . . . 6 ⊢ 2 ∈ ℝ | |
8 | 2lt7 11828 | . . . . . 6 ⊢ 2 < 7 | |
9 | 7, 8 | ltneii 10753 | . . . . 5 ⊢ 2 ≠ 7 |
10 | 9 | neii 3018 | . . . 4 ⊢ ¬ 2 = 7 |
11 | 6, 10 | pm3.2ni 877 | . . 3 ⊢ ¬ (2 = 1 ∨ 2 = 7) |
12 | 4, 11 | 2false 378 | . 2 ⊢ (0 = 1 ↔ (2 = 1 ∨ 2 = 7)) |
13 | 8nn 11733 | . . . . . 6 ⊢ 8 ∈ ℕ | |
14 | nnrp 12401 | . . . . . 6 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 8 ∈ ℝ+ |
16 | 0le2 11740 | . . . . 5 ⊢ 0 ≤ 2 | |
17 | 2lt8 11835 | . . . . 5 ⊢ 2 < 8 | |
18 | modid 13265 | . . . . 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 2903 | . . 3 ⊢ ((2 mod 8) ∈ {1, 7} ↔ 2 ∈ {1, 7}) |
21 | 2ex 11715 | . . . 4 ⊢ 2 ∈ V | |
22 | 21 | elpr 4590 | . . 3 ⊢ (2 ∈ {1, 7} ↔ (2 = 1 ∨ 2 = 7)) |
23 | 20, 22 | bitr2i 278 | . 2 ⊢ ((2 = 1 ∨ 2 = 7) ↔ (2 mod 8) ∈ {1, 7}) |
24 | 2, 12, 23 | 3bitri 299 | 1 ⊢ ((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cpr 4569 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 < clt 10675 ≤ cle 10676 ℕcn 11638 2c2 11693 7c7 11698 8c8 11699 ℝ+crp 12390 mod cmo 13238 /L clgs 25870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-prm 16016 df-phi 16103 df-pc 16174 df-lgs 25871 |
This theorem is referenced by: 2lgs 25983 |
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