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Mirrors > Home > MPE Home > Th. List > Mathboxes > 41prothprmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 41prothprm 46950. (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 12705 | . . . . . 6 ⊢ ;41 = ((;10 · 4) + 1) | |
3 | 1, 2 | eqtri 2756 | . . . . 5 ⊢ 𝑃 = ((;10 · 4) + 1) |
4 | 3 | oveq1i 7425 | . . . 4 ⊢ (𝑃 − 1) = (((;10 · 4) + 1) − 1) |
5 | 10nn 12718 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
6 | 5 | nncni 12247 | . . . . . 6 ⊢ ;10 ∈ ℂ |
7 | 4cn 12322 | . . . . . 6 ⊢ 4 ∈ ℂ | |
8 | 6, 7 | mulcli 11246 | . . . . 5 ⊢ (;10 · 4) ∈ ℂ |
9 | pncan1 11663 | . . . . 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 2756 | . . 3 ⊢ (𝑃 − 1) = (;10 · 4) |
12 | 11 | oveq1i 7425 | . 2 ⊢ ((𝑃 − 1) / 2) = ((;10 · 4) / 2) |
13 | 2cn 12312 | . . . 4 ⊢ 2 ∈ ℂ | |
14 | 2ne0 12341 | . . . 4 ⊢ 2 ≠ 0 | |
15 | 6, 7, 13, 14 | divassi 11995 | . . 3 ⊢ ((;10 · 4) / 2) = (;10 · (4 / 2)) |
16 | 4d2e2 12407 | . . . . 5 ⊢ (4 / 2) = 2 | |
17 | 16 | oveq2i 7426 | . . . 4 ⊢ (;10 · (4 / 2)) = (;10 · 2) |
18 | 2nn0 12514 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
19 | 18 | dec0u 12723 | . . . 4 ⊢ (;10 · 2) = ;20 |
20 | 17, 19 | eqtri 2756 | . . 3 ⊢ (;10 · (4 / 2)) = ;20 |
21 | 15, 20 | eqtri 2756 | . 2 ⊢ ((;10 · 4) / 2) = ;20 |
22 | 12, 21 | eqtri 2756 | 1 ⊢ ((𝑃 − 1) / 2) = ;20 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7415 ℂcc 11131 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 − cmin 11469 / cdiv 11896 2c2 12292 4c4 12294 ;cdc 12702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-dec 12703 |
This theorem is referenced by: 41prothprmlem2 46949 |
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