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Mirrors > Home > MPE Home > Th. List > 2lgslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2lgs 27466: special case of 2lgs 27466 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 27464 | . . 3 ⊢ (2 /L 2) = 0 | |
2 | 1 | eqeq1i 2740 | . 2 ⊢ ((2 /L 2) = 1 ↔ 0 = 1) |
3 | 0ne1 12335 | . . . 4 ⊢ 0 ≠ 1 | |
4 | 3 | neii 2940 | . . 3 ⊢ ¬ 0 = 1 |
5 | 1ne2 12472 | . . . . 5 ⊢ 1 ≠ 2 | |
6 | 5 | nesymi 2996 | . . . 4 ⊢ ¬ 2 = 1 |
7 | 2re 12338 | . . . . . 6 ⊢ 2 ∈ ℝ | |
8 | 2lt7 12454 | . . . . . 6 ⊢ 2 < 7 | |
9 | 7, 8 | ltneii 11372 | . . . . 5 ⊢ 2 ≠ 7 |
10 | 9 | neii 2940 | . . . 4 ⊢ ¬ 2 = 7 |
11 | 6, 10 | pm3.2ni 880 | . . 3 ⊢ ¬ (2 = 1 ∨ 2 = 7) |
12 | 4, 11 | 2false 375 | . 2 ⊢ (0 = 1 ↔ (2 = 1 ∨ 2 = 7)) |
13 | 8nn 12359 | . . . . . 6 ⊢ 8 ∈ ℕ | |
14 | nnrp 13044 | . . . . . 6 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 8 ∈ ℝ+ |
16 | 0le2 12366 | . . . . 5 ⊢ 0 ≤ 2 | |
17 | 2lt8 12461 | . . . . 5 ⊢ 2 < 8 | |
18 | modid 13933 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 8 ∈ ℝ+) ∧ (0 ≤ 2 ∧ 2 < 8)) → (2 mod 8) = 2) | |
19 | 7, 15, 16, 17, 18 | mp4an 693 | . . . 4 ⊢ (2 mod 8) = 2 |
20 | 19 | eleq1i 2830 | . . 3 ⊢ ((2 mod 8) ∈ {1, 7} ↔ 2 ∈ {1, 7}) |
21 | 2ex 12341 | . . . 4 ⊢ 2 ∈ V | |
22 | 21 | elpr 4655 | . . 3 ⊢ (2 ∈ {1, 7} ↔ (2 = 1 ∨ 2 = 7)) |
23 | 20, 22 | bitr2i 276 | . 2 ⊢ ((2 = 1 ∨ 2 = 7) ↔ (2 mod 8) ∈ {1, 7}) |
24 | 2, 12, 23 | 3bitri 297 | 1 ⊢ ((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1537 ∈ wcel 2106 {cpr 4633 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 < clt 11293 ≤ cle 11294 ℕcn 12264 2c2 12319 7c7 12324 8c8 12325 ℝ+crp 13032 mod cmo 13906 /L clgs 27353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-phi 16800 df-pc 16871 df-lgs 27354 |
This theorem is referenced by: 2lgs 27466 |
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