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Mirrors > Home > MPE Home > Th. List > decsubi | Structured version Visualization version GIF version |
Description: Difference between a numeral 𝑀 and a nonnegative integer 𝑁 (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decaddi.1 | ⊢ 𝐴 ∈ ℕ0 |
decaddi.2 | ⊢ 𝐵 ∈ ℕ0 |
decaddi.3 | ⊢ 𝑁 ∈ ℕ0 |
decaddi.4 | ⊢ 𝑀 = ;𝐴𝐵 |
decaddci.5 | ⊢ (𝐴 + 1) = 𝐷 |
decsubi.5 | ⊢ (𝐵 − 𝑁) = 𝐶 |
Ref | Expression |
---|---|
decsubi | ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12311 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
2 | decaddi.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | nn0mulcli 12128 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
4 | 3 | nn0cni 12102 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
5 | decaddi.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 12102 | . . 3 ⊢ 𝐵 ∈ ℂ |
7 | decaddi.3 | . . . 4 ⊢ 𝑁 ∈ ℕ0 | |
8 | 7 | nn0cni 12102 | . . 3 ⊢ 𝑁 ∈ ℂ |
9 | 4, 6, 8 | addsubassi 11169 | . 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
10 | decaddi.4 | . . . 4 ⊢ 𝑀 = ;𝐴𝐵 | |
11 | dfdec10 12296 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
12 | 10, 11 | eqtri 2765 | . . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵) |
13 | 12 | oveq1i 7223 | . 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁) |
14 | dfdec10 12296 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
15 | decsubi.5 | . . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶 | |
16 | 15 | eqcomi 2746 | . . . 4 ⊢ 𝐶 = (𝐵 − 𝑁) |
17 | 16 | oveq2i 7224 | . . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
18 | 14, 17 | eqtri 2765 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
19 | 9, 13, 18 | 3eqtr4i 2775 | 1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 (class class class)co 7213 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 − cmin 11062 ℕ0cn0 12090 ;cdc 12293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-sub 11064 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-dec 12294 |
This theorem is referenced by: fmtno5 44682 m5prm 44723 m7prm 44725 m11nprm 44726 341fppr2 44859 ackval41 45714 |
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