Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3dvdsdecOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of 3dvdsdec 14892 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 11371 | . . . 4 ⊢ ;𝐴𝐵 = ((10 · 𝐴) + 𝐵) | |
2 | df-10OLD 10964 | . . . . . . 7 ⊢ 10 = (9 + 1) | |
3 | 2 | oveq1i 6559 | . . . . . 6 ⊢ (10 · 𝐴) = ((9 + 1) · 𝐴) |
4 | 9cn 10985 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
5 | ax-1cn 9873 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
6 | 3dvdsdec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
7 | 6 | nn0cni 11181 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
8 | 4, 5, 7 | adddiri 9930 | . . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴)) |
9 | 7 | mulid2i 9922 | . . . . . . 7 ⊢ (1 · 𝐴) = 𝐴 |
10 | 9 | oveq2i 6560 | . . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴) |
11 | 3, 8, 10 | 3eqtri 2636 | . . . . 5 ⊢ (10 · 𝐴) = ((9 · 𝐴) + 𝐴) |
12 | 11 | oveq1i 6559 | . . . 4 ⊢ ((10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵) |
13 | 4, 7 | mulcli 9924 | . . . . 5 ⊢ (9 · 𝐴) ∈ ℂ |
14 | 3dvdsdec.b | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
15 | 14 | nn0cni 11181 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
16 | 13, 7, 15 | addassi 9927 | . . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵)) |
17 | 1, 12, 16 | 3eqtri 2636 | . . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵)) |
18 | 17 | breq2i 4591 | . 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
19 | 3z 11287 | . . 3 ⊢ 3 ∈ ℤ | |
20 | 6 | nn0zi 11279 | . . . 4 ⊢ 𝐴 ∈ ℤ |
21 | 14 | nn0zi 11279 | . . . 4 ⊢ 𝐵 ∈ ℤ |
22 | zaddcl 11294 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
23 | 20, 21, 22 | mp2an 704 | . . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ |
24 | 9nn 11069 | . . . . . 6 ⊢ 9 ∈ ℕ | |
25 | 24 | nnzi 11278 | . . . . 5 ⊢ 9 ∈ ℤ |
26 | zmulcl 11303 | . . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ) | |
27 | 25, 20, 26 | mp2an 704 | . . . 4 ⊢ (9 · 𝐴) ∈ ℤ |
28 | zmulcl 11303 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ) | |
29 | 19, 20, 28 | mp2an 704 | . . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ |
30 | dvdsmul1 14841 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴))) | |
31 | 19, 29, 30 | mp2an 704 | . . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴)) |
32 | 3t3e9 11057 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
33 | 32 | eqcomi 2619 | . . . . . . 7 ⊢ 9 = (3 · 3) |
34 | 33 | oveq1i 6559 | . . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴) |
35 | 3cn 10972 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
36 | 35, 35, 7 | mulassi 9928 | . . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴)) |
37 | 34, 36 | eqtri 2632 | . . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴)) |
38 | 31, 37 | breqtrri 4610 | . . . 4 ⊢ 3 ∥ (9 · 𝐴) |
39 | 27, 38 | pm3.2i 470 | . . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴)) |
40 | dvdsadd2b 14866 | . . 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 1416 | . 2 ⊢ (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
42 | 18, 41 | bitr4i 266 | 1 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 1c1 9816 + caddc 9818 · cmul 9820 3c3 10948 9c9 10954 10c10 10955 ℕ0cn0 11169 ℤcz 11254 ;cdc 11369 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-10OLD 10964 df-n0 11170 df-z 11255 df-dec 11370 df-dvds 14822 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |