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Mirrors > Home > MPE Home > Th. List > 3dvdsdecOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of 3dvdsdec 15054 as of 8-Sep-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
3dvdsdec.a | ⊢ 𝐴 ∈ ℕ0 |
3dvdsdec.b | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
3dvdsdecOLD | ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdecOLD 11495 | . . . 4 ⊢ ;𝐴𝐵 = ((10 · 𝐴) + 𝐵) | |
2 | df-10OLD 11087 | . . . . . . 7 ⊢ 10 = (9 + 1) | |
3 | 2 | oveq1i 6660 | . . . . . 6 ⊢ (10 · 𝐴) = ((9 + 1) · 𝐴) |
4 | 9cn 11108 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
5 | ax-1cn 9994 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
6 | 3dvdsdec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
7 | 6 | nn0cni 11304 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
8 | 4, 5, 7 | adddiri 10051 | . . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴)) |
9 | 7 | mulid2i 10043 | . . . . . . 7 ⊢ (1 · 𝐴) = 𝐴 |
10 | 9 | oveq2i 6661 | . . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴) |
11 | 3, 8, 10 | 3eqtri 2648 | . . . . 5 ⊢ (10 · 𝐴) = ((9 · 𝐴) + 𝐴) |
12 | 11 | oveq1i 6660 | . . . 4 ⊢ ((10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵) |
13 | 4, 7 | mulcli 10045 | . . . . 5 ⊢ (9 · 𝐴) ∈ ℂ |
14 | 3dvdsdec.b | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
15 | 14 | nn0cni 11304 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
16 | 13, 7, 15 | addassi 10048 | . . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵)) |
17 | 1, 12, 16 | 3eqtri 2648 | . . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵)) |
18 | 17 | breq2i 4661 | . 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
19 | 3z 11410 | . . 3 ⊢ 3 ∈ ℤ | |
20 | 6 | nn0zi 11402 | . . . 4 ⊢ 𝐴 ∈ ℤ |
21 | 14 | nn0zi 11402 | . . . 4 ⊢ 𝐵 ∈ ℤ |
22 | zaddcl 11417 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
23 | 20, 21, 22 | mp2an 708 | . . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ |
24 | 9nn 11192 | . . . . . 6 ⊢ 9 ∈ ℕ | |
25 | 24 | nnzi 11401 | . . . . 5 ⊢ 9 ∈ ℤ |
26 | zmulcl 11426 | . . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ) | |
27 | 25, 20, 26 | mp2an 708 | . . . 4 ⊢ (9 · 𝐴) ∈ ℤ |
28 | zmulcl 11426 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ) | |
29 | 19, 20, 28 | mp2an 708 | . . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ |
30 | dvdsmul1 15003 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴))) | |
31 | 19, 29, 30 | mp2an 708 | . . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴)) |
32 | 3t3e9 11180 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
33 | 32 | eqcomi 2631 | . . . . . . 7 ⊢ 9 = (3 · 3) |
34 | 33 | oveq1i 6660 | . . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴) |
35 | 3cn 11095 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
36 | 35, 35, 7 | mulassi 10049 | . . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴)) |
37 | 34, 36 | eqtri 2644 | . . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴)) |
38 | 31, 37 | breqtrri 4680 | . . . 4 ⊢ 3 ∥ (9 · 𝐴) |
39 | 27, 38 | pm3.2i 471 | . . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴)) |
40 | dvdsadd2b 15028 | . . 3 ⊢ ((3 ∈ ℤ ∧ (𝐴 + 𝐵) ∈ ℤ ∧ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴))) → (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵)))) | |
41 | 19, 23, 39, 40 | mp3an 1424 | . 2 ⊢ (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
42 | 18, 41 | bitr4i 267 | 1 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 1c1 9937 + caddc 9939 · cmul 9941 3c3 11071 9c9 11077 10c10 11078 ℕ0cn0 11292 ℤcz 11377 ;cdc 11493 ∥ cdvds 14983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-10OLD 11087 df-n0 11293 df-z 11378 df-dec 11494 df-dvds 14984 |
This theorem is referenced by: (None) |
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