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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5fac 45848. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5faclem2 | ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 12441 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 7nn0 12442 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
3 | 1, 2 | deccl 12640 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
4 | 0nn0 12435 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
5 | 3, 4 | deccl 12640 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
6 | 5, 4 | deccl 12640 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
7 | 4nn0 12439 | . . . 4 ⊢ 4 ∈ ℕ0 | |
8 | 6, 7 | deccl 12640 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 12436 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12640 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | eqid 2737 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
12 | 2nn0 12437 | . 2 ⊢ 2 ∈ ℕ0 | |
13 | 7, 4 | deccl 12640 | . . . . . . 7 ⊢ ;40 ∈ ℕ0 |
14 | 13, 12 | deccl 12640 | . . . . . 6 ⊢ ;;402 ∈ ℕ0 |
15 | 14, 4 | deccl 12640 | . . . . 5 ⊢ ;;;4020 ∈ ℕ0 |
16 | 15, 12 | deccl 12640 | . . . 4 ⊢ ;;;;40202 ∈ ℕ0 |
17 | 16, 7 | deccl 12640 | . . 3 ⊢ ;;;;;402024 ∈ ℕ0 |
18 | eqid 2737 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
19 | eqid 2737 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
20 | eqid 2737 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
21 | eqid 2737 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
22 | eqid 2737 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
23 | 3nn0 12438 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
24 | 6t6e36 12733 | . . . . . . . . . 10 ⊢ (6 · 6) = ;36 | |
25 | 3p1e4 12305 | . . . . . . . . . 10 ⊢ (3 + 1) = 4 | |
26 | 6p4e10 12697 | . . . . . . . . . 10 ⊢ (6 + 4) = ;10 | |
27 | 23, 1, 7, 24, 25, 26 | decaddci2 12687 | . . . . . . . . 9 ⊢ ((6 · 6) + 4) = ;40 |
28 | 7t6e42 12738 | . . . . . . . . 9 ⊢ (7 · 6) = ;42 | |
29 | 1, 1, 2, 22, 12, 7, 27, 28 | decmul1c 12690 | . . . . . . . 8 ⊢ (;67 · 6) = ;;402 |
30 | 6cn 12251 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
31 | 30 | mul02i 11351 | . . . . . . . 8 ⊢ (0 · 6) = 0 |
32 | 1, 3, 4, 21, 29, 31 | decmul1 12689 | . . . . . . 7 ⊢ (;;670 · 6) = ;;;4020 |
33 | 1, 5, 4, 20, 32, 31 | decmul1 12689 | . . . . . 6 ⊢ (;;;6700 · 6) = ;;;;40200 |
34 | 2cn 12235 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
35 | 34 | addid2i 11350 | . . . . . 6 ⊢ (0 + 2) = 2 |
36 | 15, 4, 12, 33, 35 | decaddi 12685 | . . . . 5 ⊢ ((;;;6700 · 6) + 2) = ;;;;40202 |
37 | 4cn 12245 | . . . . . 6 ⊢ 4 ∈ ℂ | |
38 | 6t4e24 12731 | . . . . . 6 ⊢ (6 · 4) = ;24 | |
39 | 30, 37, 38 | mulcomli 11171 | . . . . 5 ⊢ (4 · 6) = ;24 |
40 | 1, 6, 7, 19, 7, 12, 36, 39 | decmul1c 12690 | . . . 4 ⊢ (;;;;67004 · 6) = ;;;;;402024 |
41 | 30 | mulid2i 11167 | . . . 4 ⊢ (1 · 6) = 6 |
42 | 1, 8, 9, 18, 40, 41 | decmul1 12689 | . . 3 ⊢ (;;;;;670041 · 6) = ;;;;;;4020246 |
43 | eqid 2737 | . . . 4 ⊢ ;;;;;402024 = ;;;;;402024 | |
44 | 4p1e5 12306 | . . . 4 ⊢ (4 + 1) = 5 | |
45 | 16, 7, 9, 43, 44 | decaddi 12685 | . . 3 ⊢ (;;;;;402024 + 1) = ;;;;;402025 |
46 | 17, 1, 7, 42, 45, 26 | decaddci2 12687 | . 2 ⊢ ((;;;;;670041 · 6) + 4) = ;;;;;;4020250 |
47 | 1, 10, 2, 11, 12, 7, 46, 28 | decmul1c 12690 | 1 ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7362 0cc0 11058 1c1 11059 · cmul 11063 2c2 12215 3c3 12216 4c4 12217 5c5 12218 6c6 12219 7c7 12220 ;cdc 12625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-ltxr 11201 df-sub 11394 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-dec 12626 |
This theorem is referenced by: fmtno5fac 45848 |
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