![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 41prothprmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 41prothprm 45897. (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 12626 | . . . . . 6 ⊢ ;41 = ((;10 · 4) + 1) | |
3 | 1, 2 | eqtri 2761 | . . . . 5 ⊢ 𝑃 = ((;10 · 4) + 1) |
4 | 3 | oveq1i 7368 | . . . 4 ⊢ (𝑃 − 1) = (((;10 · 4) + 1) − 1) |
5 | 10nn 12639 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
6 | 5 | nncni 12168 | . . . . . 6 ⊢ ;10 ∈ ℂ |
7 | 4cn 12243 | . . . . . 6 ⊢ 4 ∈ ℂ | |
8 | 6, 7 | mulcli 11167 | . . . . 5 ⊢ (;10 · 4) ∈ ℂ |
9 | pncan1 11584 | . . . . 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 2761 | . . 3 ⊢ (𝑃 − 1) = (;10 · 4) |
12 | 11 | oveq1i 7368 | . 2 ⊢ ((𝑃 − 1) / 2) = ((;10 · 4) / 2) |
13 | 2cn 12233 | . . . 4 ⊢ 2 ∈ ℂ | |
14 | 2ne0 12262 | . . . 4 ⊢ 2 ≠ 0 | |
15 | 6, 7, 13, 14 | divassi 11916 | . . 3 ⊢ ((;10 · 4) / 2) = (;10 · (4 / 2)) |
16 | 4d2e2 12328 | . . . . 5 ⊢ (4 / 2) = 2 | |
17 | 16 | oveq2i 7369 | . . . 4 ⊢ (;10 · (4 / 2)) = (;10 · 2) |
18 | 2nn0 12435 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
19 | 18 | dec0u 12644 | . . . 4 ⊢ (;10 · 2) = ;20 |
20 | 17, 19 | eqtri 2761 | . . 3 ⊢ (;10 · (4 / 2)) = ;20 |
21 | 15, 20 | eqtri 2761 | . 2 ⊢ ((;10 · 4) / 2) = ;20 |
22 | 12, 21 | eqtri 2761 | 1 ⊢ ((𝑃 − 1) / 2) = ;20 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7358 ℂcc 11054 0cc0 11056 1c1 11057 + caddc 11059 · cmul 11061 − cmin 11390 / cdiv 11817 2c2 12213 4c4 12215 ;cdc 12623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-dec 12624 |
This theorem is referenced by: 41prothprmlem2 45896 |
Copyright terms: Public domain | W3C validator |