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Mirrors > Home > MPE Home > Th. List > 3decOLD | Structured version Visualization version GIF version |
Description: Old version of 3dec 12912. Obsolete as of 1-Aug-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
3decOLD.a | ⊢ 𝐴 ∈ ℕ0 |
3decOLD.b | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
3decOLD | ⊢ ;;𝐴𝐵𝐶 = ((((10↑2) · 𝐴) + (10 · 𝐵)) + 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdecOLD 11371 | . 2 ⊢ ;;𝐴𝐵𝐶 = ((10 · ;𝐴𝐵) + 𝐶) | |
2 | dfdecOLD 11371 | . . . . . 6 ⊢ ;𝐴𝐵 = ((10 · 𝐴) + 𝐵) | |
3 | 2 | oveq2i 6560 | . . . . 5 ⊢ (10 · ;𝐴𝐵) = (10 · ((10 · 𝐴) + 𝐵)) |
4 | 10nnOLD 11070 | . . . . . . 7 ⊢ 10 ∈ ℕ | |
5 | 4 | nncni 10907 | . . . . . 6 ⊢ 10 ∈ ℂ |
6 | 3decOLD.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
7 | 6 | nn0cni 11181 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
8 | 5, 7 | mulcli 9924 | . . . . . 6 ⊢ (10 · 𝐴) ∈ ℂ |
9 | 3decOLD.b | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
10 | 9 | nn0cni 11181 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
11 | 5, 8, 10 | adddii 9929 | . . . . 5 ⊢ (10 · ((10 · 𝐴) + 𝐵)) = ((10 · (10 · 𝐴)) + (10 · 𝐵)) |
12 | 3, 11 | eqtri 2632 | . . . 4 ⊢ (10 · ;𝐴𝐵) = ((10 · (10 · 𝐴)) + (10 · 𝐵)) |
13 | 5, 5, 7 | mulassi 9928 | . . . . . . 7 ⊢ ((10 · 10) · 𝐴) = (10 · (10 · 𝐴)) |
14 | 13 | eqcomi 2619 | . . . . . 6 ⊢ (10 · (10 · 𝐴)) = ((10 · 10) · 𝐴) |
15 | 5 | sqvali 12805 | . . . . . . . 8 ⊢ (10↑2) = (10 · 10) |
16 | 15 | eqcomi 2619 | . . . . . . 7 ⊢ (10 · 10) = (10↑2) |
17 | 16 | oveq1i 6559 | . . . . . 6 ⊢ ((10 · 10) · 𝐴) = ((10↑2) · 𝐴) |
18 | 14, 17 | eqtri 2632 | . . . . 5 ⊢ (10 · (10 · 𝐴)) = ((10↑2) · 𝐴) |
19 | 18 | oveq1i 6559 | . . . 4 ⊢ ((10 · (10 · 𝐴)) + (10 · 𝐵)) = (((10↑2) · 𝐴) + (10 · 𝐵)) |
20 | 12, 19 | eqtri 2632 | . . 3 ⊢ (10 · ;𝐴𝐵) = (((10↑2) · 𝐴) + (10 · 𝐵)) |
21 | 20 | oveq1i 6559 | . 2 ⊢ ((10 · ;𝐴𝐵) + 𝐶) = ((((10↑2) · 𝐴) + (10 · 𝐵)) + 𝐶) |
22 | 1, 21 | eqtri 2632 | 1 ⊢ ;;𝐴𝐵𝐶 = ((((10↑2) · 𝐴) + (10 · 𝐵)) + 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 + caddc 9818 · cmul 9820 2c2 10947 10c10 10955 ℕ0cn0 11169 ;cdc 11369 ↑cexp 12722 |
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-cnex 9871 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-2nd 7060 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-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: 3dvds2decOLD 14895 |
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