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Mirrors > Home > MPE Home > Th. List > Mathboxes > 235t711 | Structured version Visualization version GIF version |
Description: Calculate a product by
long multiplication as a base comparison with other
multiplication algorithms.
Conveniently, 711 has two ones which greatly simplifies calculations like 235 · 1. There isn't a higher level mulcomli 11273 saving the lower level uses of mulcomli 11273 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12838 are added then this proof would benefit more than ex-decpmul 42107. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12399 or 8t7e56 12849. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) |
Ref | Expression |
---|---|
235t711 | ⊢ (;;235 · ;;711) = ;;;;;167085 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 12541 | . . . 4 ⊢ 2 ∈ ℕ0 | |
2 | 3nn0 12542 | . . . 4 ⊢ 3 ∈ ℕ0 | |
3 | 1, 2 | deccl 12744 | . . 3 ⊢ ;23 ∈ ℕ0 |
4 | 5nn0 12544 | . . 3 ⊢ 5 ∈ ℕ0 | |
5 | 3, 4 | deccl 12744 | . 2 ⊢ ;;235 ∈ ℕ0 |
6 | 7nn0 12546 | . . 3 ⊢ 7 ∈ ℕ0 | |
7 | 1nn0 12540 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 6, 7 | deccl 12744 | . 2 ⊢ ;71 ∈ ℕ0 |
9 | eqid 2726 | . 2 ⊢ ;;711 = ;;711 | |
10 | eqid 2726 | . . 3 ⊢ ;71 = ;71 | |
11 | eqid 2726 | . . 3 ⊢ ;23 = ;23 | |
12 | 8nn0 12547 | . . 3 ⊢ 8 ∈ ℕ0 | |
13 | eqid 2726 | . . . 4 ⊢ ;;235 = ;;235 | |
14 | 3 | nn0cni 12536 | . . . . 5 ⊢ ;23 ∈ ℂ |
15 | 2cn 12339 | . . . . 5 ⊢ 2 ∈ ℂ | |
16 | 3p2e5 12415 | . . . . . 6 ⊢ (3 + 2) = 5 | |
17 | 1, 2, 1, 11, 16 | decaddi 12789 | . . . . 5 ⊢ (;23 + 2) = ;25 |
18 | 14, 15, 17 | addcomli 11456 | . . . 4 ⊢ (2 + ;23) = ;25 |
19 | 0nn0 12539 | . . . 4 ⊢ 0 ∈ ℕ0 | |
20 | 4nn0 12543 | . . . 4 ⊢ 4 ∈ ℕ0 | |
21 | 6nn0 12545 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
22 | 7, 21 | deccl 12744 | . . . . 5 ⊢ ;16 ∈ ℕ0 |
23 | 1, 20 | nn0addcli 12561 | . . . . 5 ⊢ (2 + 4) ∈ ℕ0 |
24 | 7cn 12358 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
25 | 7t2e14 12838 | . . . . . . . 8 ⊢ (7 · 2) = ;14 | |
26 | 24, 15, 25 | mulcomli 11273 | . . . . . . 7 ⊢ (2 · 7) = ;14 |
27 | 4p2e6 12417 | . . . . . . 7 ⊢ (4 + 2) = 6 | |
28 | 7, 20, 1, 26, 27 | decaddi 12789 | . . . . . 6 ⊢ ((2 · 7) + 2) = ;16 |
29 | 3cn 12345 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
30 | 7t3e21 12839 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
31 | 24, 29, 30 | mulcomli 11273 | . . . . . 6 ⊢ (3 · 7) = ;21 |
32 | 6, 1, 2, 11, 7, 1, 28, 31 | decmul1c 12794 | . . . . 5 ⊢ (;23 · 7) = ;;161 |
33 | 4cn 12349 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
34 | 15, 33 | addcli 11270 | . . . . . 6 ⊢ (2 + 4) ∈ ℂ |
35 | ax-1cn 11216 | . . . . . 6 ⊢ 1 ∈ ℂ | |
36 | 33, 15, 27 | addcomli 11456 | . . . . . . . 8 ⊢ (2 + 4) = 6 |
37 | 36 | oveq1i 7434 | . . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1) |
38 | 6p1e7 12412 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
39 | 37, 38 | eqtri 2754 | . . . . . 6 ⊢ ((2 + 4) + 1) = 7 |
40 | 34, 35, 39 | addcomli 11456 | . . . . 5 ⊢ (1 + (2 + 4)) = 7 |
41 | 22, 7, 23, 32, 40 | decaddi 12789 | . . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167 |
42 | 5cn 12352 | . . . . . 6 ⊢ 5 ∈ ℂ | |
43 | 7t5e35 12841 | . . . . . 6 ⊢ (7 · 5) = ;35 | |
44 | 24, 42, 43 | mulcomli 11273 | . . . . 5 ⊢ (5 · 7) = ;35 |
45 | 3p1e4 12409 | . . . . 5 ⊢ (3 + 1) = 4 | |
46 | 5p5e10 12800 | . . . . 5 ⊢ (5 + 5) = ;10 | |
47 | 2, 4, 4, 44, 45, 46 | decaddci2 12791 | . . . 4 ⊢ ((5 · 7) + 5) = ;40 |
48 | 3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47 | decmac 12781 | . . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670 |
49 | 5 | nn0cni 12536 | . . . . 5 ⊢ ;;235 ∈ ℂ |
50 | 49 | mulridi 11268 | . . . 4 ⊢ (;;235 · 1) = ;;235 |
51 | 5p3e8 12421 | . . . 4 ⊢ (5 + 3) = 8 | |
52 | 3, 4, 2, 50, 51 | decaddi 12789 | . . 3 ⊢ ((;;235 · 1) + 3) = ;;238 |
53 | 6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52 | decma2c 12782 | . 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708 |
54 | 5, 8, 7, 9, 4, 3, 53, 50 | decmul2c 12795 | 1 ⊢ (;;235 · ;;711) = ;;;;;167085 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7424 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 7c7 12324 8c8 12325 ;cdc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12730 |
This theorem is referenced by: (None) |
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