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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 12702 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
2 | decaddi.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | nn0mulcli 12517 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
4 | 3 | nn0cni 12491 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
5 | decaddi.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 12491 | . . 3 ⊢ 𝐵 ∈ ℂ |
7 | decaddi.3 | . . . 4 ⊢ 𝑁 ∈ ℕ0 | |
8 | 7 | nn0cni 12491 | . . 3 ⊢ 𝑁 ∈ ℂ |
9 | 4, 6, 8 | addsubassi 11558 | . 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
10 | decaddi.4 | . . . 4 ⊢ 𝑀 = ;𝐴𝐵 | |
11 | dfdec10 12687 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
12 | 10, 11 | eqtri 2759 | . . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵) |
13 | 12 | oveq1i 7422 | . 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁) |
14 | dfdec10 12687 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
15 | decsubi.5 | . . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶 | |
16 | 15 | eqcomi 2740 | . . . 4 ⊢ 𝐶 = (𝐵 − 𝑁) |
17 | 16 | oveq2i 7423 | . . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
18 | 14, 17 | eqtri 2759 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
19 | 9, 13, 18 | 3eqtr4i 2769 | 1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 (class class class)co 7412 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 − cmin 11451 ℕ0cn0 12479 ;cdc 12684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-ltxr 11260 df-sub 11453 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-dec 12685 |
This theorem is referenced by: fmtno5 46684 m5prm 46725 m7prm 46727 m11nprm 46728 341fppr2 46861 ackval41 47543 |
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