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Mirrors > Home > MPE Home > Th. List > decmul1 | Structured version Visualization version GIF version |
Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) Remove hypothesis 𝐷 ∈ ℕ0. (Revised by Steven Nguyen, 7-Dec-2022.) |
Ref | Expression |
---|---|
decmul1.p | ⊢ 𝑃 ∈ ℕ0 |
decmul1.a | ⊢ 𝐴 ∈ ℕ0 |
decmul1.b | ⊢ 𝐵 ∈ ℕ0 |
decmul1.n | ⊢ 𝑁 = ;𝐴𝐵 |
decmul1.c | ⊢ (𝐴 · 𝑃) = 𝐶 |
decmul1.d | ⊢ (𝐵 · 𝑃) = 𝐷 |
Ref | Expression |
---|---|
decmul1 | ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decmul1.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
2 | decmul1.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | decmul1.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
4 | 2, 3 | deccl 12116 | . . . 4 ⊢ ;𝐴𝐵 ∈ ℕ0 |
5 | 1, 4 | eqeltri 2912 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
6 | decmul1.p | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
7 | 5, 6 | num0u 12112 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑁 · 𝑃) + 0) |
8 | 0nn0 11915 | . . 3 ⊢ 0 ∈ ℕ0 | |
9 | decmul1.c | . . 3 ⊢ (𝐴 · 𝑃) = 𝐶 | |
10 | 3, 6 | nn0mulcli 11938 | . . . . . 6 ⊢ (𝐵 · 𝑃) ∈ ℕ0 |
11 | 10 | nn0cni 11912 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
12 | 11 | addid1i 10830 | . . . 4 ⊢ ((𝐵 · 𝑃) + 0) = (𝐵 · 𝑃) |
13 | decmul1.d | . . . 4 ⊢ (𝐵 · 𝑃) = 𝐷 | |
14 | 12, 13 | eqtri 2847 | . . 3 ⊢ ((𝐵 · 𝑃) + 0) = 𝐷 |
15 | 2, 3, 8, 1, 6, 9, 14 | decrmanc 12158 | . 2 ⊢ ((𝑁 · 𝑃) + 0) = ;𝐶𝐷 |
16 | 7, 15 | eqtri 2847 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 (class class class)co 7159 0cc0 10540 + caddc 10543 · cmul 10545 ℕ0cn0 11900 ;cdc 12101 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-dec 12102 |
This theorem is referenced by: 37prm 16457 1259lem3 16469 1259lem4 16470 2503lem1 16473 2503lem2 16474 4001lem1 16477 4001lem2 16478 4001lem3 16479 4001prm 16481 log2ublem3 25529 log2ub 25530 bpos1 25862 ex-prmo 28241 dpmul 30593 decpmulnc 39179 sqdeccom12 39181 ex-decpmul 39184 fmtno5lem3 43724 fmtno4prmfac193 43742 fmtno4nprmfac193 43743 fmtno5faclem1 43748 fmtno5faclem2 43749 2exp7 43769 |
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