| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfdec100 | Structured version Visualization version GIF version | ||
| Description: Split the hundreds from a decimal value. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| Ref | Expression |
|---|---|
| dfdec100.a | ⊢ 𝐴 ∈ ℕ0 |
| dfdec100.b | ⊢ 𝐵 ∈ ℕ0 |
| dfdec100.c | ⊢ 𝐶 ∈ ℝ |
| Ref | Expression |
|---|---|
| dfdec100 | ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfdec10 12736 | . . 3 ⊢ ;𝐵𝐶 = ((;10 · 𝐵) + 𝐶) | |
| 2 | 1 | oveq2i 7442 | . 2 ⊢ ((;;100 · 𝐴) + ;𝐵𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
| 3 | 10nn0 12751 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
| 4 | 3 | dec0u 12754 | . . . . 5 ⊢ (;10 · ;10) = ;;100 |
| 5 | 3 | nn0cni 12538 | . . . . . 6 ⊢ ;10 ∈ ℂ |
| 6 | 5, 5 | mulcli 11268 | . . . . 5 ⊢ (;10 · ;10) ∈ ℂ |
| 7 | 4, 6 | eqeltrri 2838 | . . . 4 ⊢ ;;100 ∈ ℂ |
| 8 | dfdec100.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
| 9 | 8 | nn0cni 12538 | . . . 4 ⊢ 𝐴 ∈ ℂ |
| 10 | 7, 9 | mulcli 11268 | . . 3 ⊢ (;;100 · 𝐴) ∈ ℂ |
| 11 | dfdec100.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
| 12 | 11 | nn0cni 12538 | . . . 4 ⊢ 𝐵 ∈ ℂ |
| 13 | 5, 12 | mulcli 11268 | . . 3 ⊢ (;10 · 𝐵) ∈ ℂ |
| 14 | dfdec100.c | . . . 4 ⊢ 𝐶 ∈ ℝ | |
| 15 | 14 | recni 11275 | . . 3 ⊢ 𝐶 ∈ ℂ |
| 16 | 10, 13, 15 | addassi 11271 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
| 17 | dfdec10 12736 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
| 18 | dfdec10 12736 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
| 19 | 18 | oveq2i 7442 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
| 20 | 5, 9 | mulcli 11268 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
| 21 | 5, 20, 12 | adddii 11273 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
| 22 | 5, 5, 9 | mulassi 11272 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
| 23 | 4 | oveq1i 7441 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
| 24 | 22, 23 | eqtr3i 2767 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = (;;100 · 𝐴) |
| 25 | 24 | oveq1i 7441 | . . . . 5 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
| 26 | 19, 21, 25 | 3eqtri 2769 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
| 27 | 26 | oveq1i 7441 | . . 3 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) |
| 28 | 17, 27 | eqtr2i 2766 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ;;𝐴𝐵𝐶 |
| 29 | 2, 16, 28 | 3eqtr2ri 2772 | 1 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕ0cn0 12526 ;cdc 12733 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 |
| This theorem is referenced by: dpmul100 32879 dpmul1000 32881 dpmul4 32896 |
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