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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 11270 saving the lower level uses of mulcomli 11270 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12842 are added then this proof would benefit more than ex-decpmul 42340. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12401 or 8t7e56 12853. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| 235t711 | ⊢ (;;235 · ;;711) = ;;;;;167085 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2nn0 12543 | . . . 4 ⊢ 2 ∈ ℕ0 | |
| 2 | 3nn0 12544 | . . . 4 ⊢ 3 ∈ ℕ0 | |
| 3 | 1, 2 | deccl 12748 | . . 3 ⊢ ;23 ∈ ℕ0 |
| 4 | 5nn0 12546 | . . 3 ⊢ 5 ∈ ℕ0 | |
| 5 | 3, 4 | deccl 12748 | . 2 ⊢ ;;235 ∈ ℕ0 |
| 6 | 7nn0 12548 | . . 3 ⊢ 7 ∈ ℕ0 | |
| 7 | 1nn0 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 8 | 6, 7 | deccl 12748 | . 2 ⊢ ;71 ∈ ℕ0 |
| 9 | eqid 2737 | . 2 ⊢ ;;711 = ;;711 | |
| 10 | eqid 2737 | . . 3 ⊢ ;71 = ;71 | |
| 11 | eqid 2737 | . . 3 ⊢ ;23 = ;23 | |
| 12 | 8nn0 12549 | . . 3 ⊢ 8 ∈ ℕ0 | |
| 13 | eqid 2737 | . . . 4 ⊢ ;;235 = ;;235 | |
| 14 | 3 | nn0cni 12538 | . . . . 5 ⊢ ;23 ∈ ℂ |
| 15 | 2cn 12341 | . . . . 5 ⊢ 2 ∈ ℂ | |
| 16 | 3p2e5 12417 | . . . . . 6 ⊢ (3 + 2) = 5 | |
| 17 | 1, 2, 1, 11, 16 | decaddi 12793 | . . . . 5 ⊢ (;23 + 2) = ;25 |
| 18 | 14, 15, 17 | addcomli 11453 | . . . 4 ⊢ (2 + ;23) = ;25 |
| 19 | 0nn0 12541 | . . . 4 ⊢ 0 ∈ ℕ0 | |
| 20 | 4nn0 12545 | . . . 4 ⊢ 4 ∈ ℕ0 | |
| 21 | 6nn0 12547 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
| 22 | 7, 21 | deccl 12748 | . . . . 5 ⊢ ;16 ∈ ℕ0 |
| 23 | 1, 20 | nn0addcli 12563 | . . . . 5 ⊢ (2 + 4) ∈ ℕ0 |
| 24 | 7cn 12360 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
| 25 | 7t2e14 12842 | . . . . . . . 8 ⊢ (7 · 2) = ;14 | |
| 26 | 24, 15, 25 | mulcomli 11270 | . . . . . . 7 ⊢ (2 · 7) = ;14 |
| 27 | 4p2e6 12419 | . . . . . . 7 ⊢ (4 + 2) = 6 | |
| 28 | 7, 20, 1, 26, 27 | decaddi 12793 | . . . . . 6 ⊢ ((2 · 7) + 2) = ;16 |
| 29 | 3cn 12347 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 30 | 7t3e21 12843 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
| 31 | 24, 29, 30 | mulcomli 11270 | . . . . . 6 ⊢ (3 · 7) = ;21 |
| 32 | 6, 1, 2, 11, 7, 1, 28, 31 | decmul1c 12798 | . . . . 5 ⊢ (;23 · 7) = ;;161 |
| 33 | 4cn 12351 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
| 34 | 15, 33 | addcli 11267 | . . . . . 6 ⊢ (2 + 4) ∈ ℂ |
| 35 | ax-1cn 11213 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 36 | 33, 15, 27 | addcomli 11453 | . . . . . . . 8 ⊢ (2 + 4) = 6 |
| 37 | 36 | oveq1i 7441 | . . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1) |
| 38 | 6p1e7 12414 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
| 39 | 37, 38 | eqtri 2765 | . . . . . 6 ⊢ ((2 + 4) + 1) = 7 |
| 40 | 34, 35, 39 | addcomli 11453 | . . . . 5 ⊢ (1 + (2 + 4)) = 7 |
| 41 | 22, 7, 23, 32, 40 | decaddi 12793 | . . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167 |
| 42 | 5cn 12354 | . . . . . 6 ⊢ 5 ∈ ℂ | |
| 43 | 7t5e35 12845 | . . . . . 6 ⊢ (7 · 5) = ;35 | |
| 44 | 24, 42, 43 | mulcomli 11270 | . . . . 5 ⊢ (5 · 7) = ;35 |
| 45 | 3p1e4 12411 | . . . . 5 ⊢ (3 + 1) = 4 | |
| 46 | 5p5e10 12804 | . . . . 5 ⊢ (5 + 5) = ;10 | |
| 47 | 2, 4, 4, 44, 45, 46 | decaddci2 12795 | . . . 4 ⊢ ((5 · 7) + 5) = ;40 |
| 48 | 3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47 | decmac 12785 | . . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670 |
| 49 | 5 | nn0cni 12538 | . . . . 5 ⊢ ;;235 ∈ ℂ |
| 50 | 49 | mulridi 11265 | . . . 4 ⊢ (;;235 · 1) = ;;235 |
| 51 | 5p3e8 12423 | . . . 4 ⊢ (5 + 3) = 8 | |
| 52 | 3, 4, 2, 50, 51 | decaddi 12793 | . . 3 ⊢ ((;;235 · 1) + 3) = ;;238 |
| 53 | 6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52 | decma2c 12786 | . 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708 |
| 54 | 5, 8, 7, 9, 4, 3, 53, 50 | decmul2c 12799 | 1 ⊢ (;;235 · ;;711) = ;;;;;167085 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 2c2 12321 3c3 12322 4c4 12323 5c5 12324 6c6 12325 7c7 12326 8c8 12327 ;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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 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-sub 11494 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: (None) |
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