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| Mirrors > Home > ILE Home > Th. List > dec2dvds | GIF version | ||
| Description: Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
| Ref | Expression |
|---|---|
| dec2dvds.1 | ⊢ 𝐴 ∈ ℕ0 |
| dec2dvds.2 | ⊢ 𝐵 ∈ ℕ0 |
| dec2dvds.3 | ⊢ (𝐵 · 2) = 𝐶 |
| dec2dvds.4 | ⊢ 𝐷 = (𝐶 + 1) |
| Ref | Expression |
|---|---|
| dec2dvds | ⊢ ¬ 2 ∥ ;𝐴𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 5nn0 9422 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
| 2 | 1 | nn0zi 9501 | . . . . . . . 8 ⊢ 5 ∈ ℤ |
| 3 | 2z 9507 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
| 4 | dvdsmul2 12377 | . . . . . . . 8 ⊢ ((5 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (5 · 2)) | |
| 5 | 2, 3, 4 | mp2an 426 | . . . . . . 7 ⊢ 2 ∥ (5 · 2) |
| 6 | 5t2e10 9710 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
| 7 | 5, 6 | breqtri 4113 | . . . . . 6 ⊢ 2 ∥ ;10 |
| 8 | 10nn0 9628 | . . . . . . . 8 ⊢ ;10 ∈ ℕ0 | |
| 9 | 8 | nn0zi 9501 | . . . . . . 7 ⊢ ;10 ∈ ℤ |
| 10 | dec2dvds.1 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
| 11 | 10 | nn0zi 9501 | . . . . . . 7 ⊢ 𝐴 ∈ ℤ |
| 12 | dvdsmultr1 12394 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ ;10 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 ∥ ;10 → 2 ∥ (;10 · 𝐴))) | |
| 13 | 3, 9, 11, 12 | mp3an 1373 | . . . . . 6 ⊢ (2 ∥ ;10 → 2 ∥ (;10 · 𝐴)) |
| 14 | 7, 13 | ax-mp 5 | . . . . 5 ⊢ 2 ∥ (;10 · 𝐴) |
| 15 | dec2dvds.2 | . . . . . . . 8 ⊢ 𝐵 ∈ ℕ0 | |
| 16 | 15 | nn0zi 9501 | . . . . . . 7 ⊢ 𝐵 ∈ ℤ |
| 17 | dvdsmul2 12377 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (𝐵 · 2)) | |
| 18 | 16, 3, 17 | mp2an 426 | . . . . . 6 ⊢ 2 ∥ (𝐵 · 2) |
| 19 | dec2dvds.3 | . . . . . 6 ⊢ (𝐵 · 2) = 𝐶 | |
| 20 | 18, 19 | breqtri 4113 | . . . . 5 ⊢ 2 ∥ 𝐶 |
| 21 | 8, 10 | nn0mulcli 9440 | . . . . . . 7 ⊢ (;10 · 𝐴) ∈ ℕ0 |
| 22 | 21 | nn0zi 9501 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℤ |
| 23 | 2nn0 9419 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
| 24 | 15, 23 | nn0mulcli 9440 | . . . . . . . 8 ⊢ (𝐵 · 2) ∈ ℕ0 |
| 25 | 19, 24 | eqeltrri 2305 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 |
| 26 | 25 | nn0zi 9501 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
| 27 | dvds2add 12388 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ (;10 · 𝐴) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶))) | |
| 28 | 3, 22, 26, 27 | mp3an 1373 | . . . . 5 ⊢ ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶)) |
| 29 | 14, 20, 28 | mp2an 426 | . . . 4 ⊢ 2 ∥ ((;10 · 𝐴) + 𝐶) |
| 30 | dfdec10 9614 | . . . 4 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
| 31 | 29, 30 | breqtrri 4115 | . . 3 ⊢ 2 ∥ ;𝐴𝐶 |
| 32 | 10, 25 | deccl 9625 | . . . . 5 ⊢ ;𝐴𝐶 ∈ ℕ0 |
| 33 | 32 | nn0zi 9501 | . . . 4 ⊢ ;𝐴𝐶 ∈ ℤ |
| 34 | 2nn 9305 | . . . 4 ⊢ 2 ∈ ℕ | |
| 35 | 1lt2 9313 | . . . 4 ⊢ 1 < 2 | |
| 36 | ndvdsp1 12495 | . . . 4 ⊢ ((;𝐴𝐶 ∈ ℤ ∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1))) | |
| 37 | 33, 34, 35, 36 | mp3an 1373 | . . 3 ⊢ (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1)) |
| 38 | 31, 37 | ax-mp 5 | . 2 ⊢ ¬ 2 ∥ (;𝐴𝐶 + 1) |
| 39 | dec2dvds.4 | . . . . 5 ⊢ 𝐷 = (𝐶 + 1) | |
| 40 | 39 | eqcomi 2235 | . . . 4 ⊢ (𝐶 + 1) = 𝐷 |
| 41 | eqid 2231 | . . . 4 ⊢ ;𝐴𝐶 = ;𝐴𝐶 | |
| 42 | 10, 25, 40, 41 | decsuc 9641 | . . 3 ⊢ (;𝐴𝐶 + 1) = ;𝐴𝐷 |
| 43 | 42 | breq2i 4096 | . 2 ⊢ (2 ∥ (;𝐴𝐶 + 1) ↔ 2 ∥ ;𝐴𝐷) |
| 44 | 38, 43 | mtbi 676 | 1 ⊢ ¬ 2 ∥ ;𝐴𝐷 |
| Colors of variables: wff set class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 (class class class)co 6018 0cc0 8032 1c1 8033 + caddc 8035 · cmul 8037 < clt 8214 ℕcn 9143 2c2 9194 5c5 9197 ℕ0cn0 9402 ℤcz 9479 ;cdc 9611 ∥ cdvds 12350 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 |
| This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-5 9205 df-6 9206 df-7 9207 df-8 9208 df-9 9209 df-n0 9403 df-z 9480 df-dec 9612 df-uz 9756 df-q 9854 df-rp 9889 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-cj 11404 df-re 11405 df-im 11406 df-rsqrt 11560 df-abs 11561 df-dvds 12351 |
| This theorem is referenced by: (None) |
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