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Mirrors > Home > MPE Home > Th. List > Mathboxes > 41prothprmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 41prothprm 42567. (Contributed by AV, 4-Jul-2020.) |
Ref | Expression |
---|---|
41prothprm.p | ⊢ 𝑃 = ;41 |
Ref | Expression |
---|---|
41prothprmlem1 | ⊢ ((𝑃 − 1) / 2) = ;20 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 41prothprm.p | . . . . . 6 ⊢ 𝑃 = ;41 | |
2 | dfdec10 11852 | . . . . . 6 ⊢ ;41 = ((;10 · 4) + 1) | |
3 | 1, 2 | eqtri 2802 | . . . . 5 ⊢ 𝑃 = ((;10 · 4) + 1) |
4 | 3 | oveq1i 6934 | . . . 4 ⊢ (𝑃 − 1) = (((;10 · 4) + 1) − 1) |
5 | 10nn 11865 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
6 | 5 | nncni 11389 | . . . . . 6 ⊢ ;10 ∈ ℂ |
7 | 4cn 11465 | . . . . . 6 ⊢ 4 ∈ ℂ | |
8 | 6, 7 | mulcli 10386 | . . . . 5 ⊢ (;10 · 4) ∈ ℂ |
9 | pncan1 10801 | . . . . 5 ⊢ ((;10 · 4) ∈ ℂ → (((;10 · 4) + 1) − 1) = (;10 · 4)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (((;10 · 4) + 1) − 1) = (;10 · 4) |
11 | 4, 10 | eqtri 2802 | . . 3 ⊢ (𝑃 − 1) = (;10 · 4) |
12 | 11 | oveq1i 6934 | . 2 ⊢ ((𝑃 − 1) / 2) = ((;10 · 4) / 2) |
13 | 2cn 11454 | . . . 4 ⊢ 2 ∈ ℂ | |
14 | 2ne0 11490 | . . . 4 ⊢ 2 ≠ 0 | |
15 | 6, 7, 13, 14 | divassi 11133 | . . 3 ⊢ ((;10 · 4) / 2) = (;10 · (4 / 2)) |
16 | 4d2e2 11556 | . . . . 5 ⊢ (4 / 2) = 2 | |
17 | 16 | oveq2i 6935 | . . . 4 ⊢ (;10 · (4 / 2)) = (;10 · 2) |
18 | 2nn0 11665 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
19 | 18 | dec0u 11871 | . . . 4 ⊢ (;10 · 2) = ;20 |
20 | 17, 19 | eqtri 2802 | . . 3 ⊢ (;10 · (4 / 2)) = ;20 |
21 | 15, 20 | eqtri 2802 | . 2 ⊢ ((;10 · 4) / 2) = ;20 |
22 | 12, 21 | eqtri 2802 | 1 ⊢ ((𝑃 − 1) / 2) = ;20 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 (class class class)co 6924 ℂcc 10272 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 / cdiv 11034 2c2 11434 4c4 11436 ;cdc 11849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-dec 11850 |
This theorem is referenced by: 41prothprmlem2 42566 |
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