Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2lgslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2lgs 26536: special case of 2lgs 26536 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 26534 | . . 3 ⊢ (2 /L 2) = 0 | |
2 | 1 | eqeq1i 2744 | . 2 ⊢ ((2 /L 2) = 1 ↔ 0 = 1) |
3 | 0ne1 12027 | . . . 4 ⊢ 0 ≠ 1 | |
4 | 3 | neii 2946 | . . 3 ⊢ ¬ 0 = 1 |
5 | 1ne2 12164 | . . . . 5 ⊢ 1 ≠ 2 | |
6 | 5 | nesymi 3002 | . . . 4 ⊢ ¬ 2 = 1 |
7 | 2re 12030 | . . . . . 6 ⊢ 2 ∈ ℝ | |
8 | 2lt7 12146 | . . . . . 6 ⊢ 2 < 7 | |
9 | 7, 8 | ltneii 11071 | . . . . 5 ⊢ 2 ≠ 7 |
10 | 9 | neii 2946 | . . . 4 ⊢ ¬ 2 = 7 |
11 | 6, 10 | pm3.2ni 877 | . . 3 ⊢ ¬ (2 = 1 ∨ 2 = 7) |
12 | 4, 11 | 2false 375 | . 2 ⊢ (0 = 1 ↔ (2 = 1 ∨ 2 = 7)) |
13 | 8nn 12051 | . . . . . 6 ⊢ 8 ∈ ℕ | |
14 | nnrp 12723 | . . . . . 6 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 8 ∈ ℝ+ |
16 | 0le2 12058 | . . . . 5 ⊢ 0 ≤ 2 | |
17 | 2lt8 12153 | . . . . 5 ⊢ 2 < 8 | |
18 | modid 13597 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 8 ∈ ℝ+) ∧ (0 ≤ 2 ∧ 2 < 8)) → (2 mod 8) = 2) | |
19 | 7, 15, 16, 17, 18 | mp4an 689 | . . . 4 ⊢ (2 mod 8) = 2 |
20 | 19 | eleq1i 2830 | . . 3 ⊢ ((2 mod 8) ∈ {1, 7} ↔ 2 ∈ {1, 7}) |
21 | 2ex 12033 | . . . 4 ⊢ 2 ∈ V | |
22 | 21 | elpr 4589 | . . 3 ⊢ (2 ∈ {1, 7} ↔ (2 = 1 ∨ 2 = 7)) |
23 | 20, 22 | bitr2i 275 | . 2 ⊢ ((2 = 1 ∨ 2 = 7) ↔ (2 mod 8) ∈ {1, 7}) |
24 | 2, 12, 23 | 3bitri 296 | 1 ⊢ ((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 843 = wceq 1541 ∈ wcel 2109 {cpr 4568 class class class wbr 5078 (class class class)co 7268 ℝcr 10854 0cc0 10855 1c1 10856 < clt 10993 ≤ cle 10994 ℕcn 11956 2c2 12011 7c7 12016 8c8 12017 ℝ+crp 12712 mod cmo 13570 /L clgs 26423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-oadd 8285 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-n0 12217 df-xnn0 12289 df-z 12303 df-uz 12565 df-q 12671 df-rp 12713 df-fz 13222 df-fzo 13365 df-fl 13493 df-mod 13571 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-dvds 15945 df-gcd 16183 df-prm 16358 df-phi 16448 df-pc 16519 df-lgs 26424 |
This theorem is referenced by: 2lgs 26536 |
Copyright terms: Public domain | W3C validator |