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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 11554 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
2 | decaddi.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | nn0mulcli 11369 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
4 | 3 | nn0cni 11342 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
5 | decaddi.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 11342 | . . 3 ⊢ 𝐵 ∈ ℂ |
7 | decaddi.3 | . . . 4 ⊢ 𝑁 ∈ ℕ0 | |
8 | 7 | nn0cni 11342 | . . 3 ⊢ 𝑁 ∈ ℂ |
9 | 4, 6, 8 | addsubassi 10410 | . 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
10 | decaddi.4 | . . . 4 ⊢ 𝑀 = ;𝐴𝐵 | |
11 | dfdec10 11535 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
12 | 10, 11 | eqtri 2673 | . . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵) |
13 | 12 | oveq1i 6700 | . 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁) |
14 | dfdec10 11535 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
15 | decsubi.5 | . . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶 | |
16 | 15 | eqcomi 2660 | . . . 4 ⊢ 𝐶 = (𝐵 − 𝑁) |
17 | 16 | oveq2i 6701 | . . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
18 | 14, 17 | eqtri 2673 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
19 | 9, 13, 18 | 3eqtr4i 2683 | 1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 (class class class)co 6690 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 − cmin 10304 ℕ0cn0 11330 ;cdc 11531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-dec 11532 |
This theorem is referenced by: fmtno5 41794 m5prm 41838 m7prm 41841 m11nprm 41843 |
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