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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5fac 46981. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5faclem2 | ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 12518 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 7nn0 12519 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
3 | 1, 2 | deccl 12717 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
4 | 0nn0 12512 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
5 | 3, 4 | deccl 12717 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
6 | 5, 4 | deccl 12717 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
7 | 4nn0 12516 | . . . 4 ⊢ 4 ∈ ℕ0 | |
8 | 6, 7 | deccl 12717 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 12513 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12717 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | eqid 2725 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
12 | 2nn0 12514 | . 2 ⊢ 2 ∈ ℕ0 | |
13 | 7, 4 | deccl 12717 | . . . . . . 7 ⊢ ;40 ∈ ℕ0 |
14 | 13, 12 | deccl 12717 | . . . . . 6 ⊢ ;;402 ∈ ℕ0 |
15 | 14, 4 | deccl 12717 | . . . . 5 ⊢ ;;;4020 ∈ ℕ0 |
16 | 15, 12 | deccl 12717 | . . . 4 ⊢ ;;;;40202 ∈ ℕ0 |
17 | 16, 7 | deccl 12717 | . . 3 ⊢ ;;;;;402024 ∈ ℕ0 |
18 | eqid 2725 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
19 | eqid 2725 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
20 | eqid 2725 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
21 | eqid 2725 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
22 | eqid 2725 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
23 | 3nn0 12515 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
24 | 6t6e36 12810 | . . . . . . . . . 10 ⊢ (6 · 6) = ;36 | |
25 | 3p1e4 12382 | . . . . . . . . . 10 ⊢ (3 + 1) = 4 | |
26 | 6p4e10 12774 | . . . . . . . . . 10 ⊢ (6 + 4) = ;10 | |
27 | 23, 1, 7, 24, 25, 26 | decaddci2 12764 | . . . . . . . . 9 ⊢ ((6 · 6) + 4) = ;40 |
28 | 7t6e42 12815 | . . . . . . . . 9 ⊢ (7 · 6) = ;42 | |
29 | 1, 1, 2, 22, 12, 7, 27, 28 | decmul1c 12767 | . . . . . . . 8 ⊢ (;67 · 6) = ;;402 |
30 | 6cn 12328 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
31 | 30 | mul02i 11428 | . . . . . . . 8 ⊢ (0 · 6) = 0 |
32 | 1, 3, 4, 21, 29, 31 | decmul1 12766 | . . . . . . 7 ⊢ (;;670 · 6) = ;;;4020 |
33 | 1, 5, 4, 20, 32, 31 | decmul1 12766 | . . . . . 6 ⊢ (;;;6700 · 6) = ;;;;40200 |
34 | 2cn 12312 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
35 | 34 | addlidi 11427 | . . . . . 6 ⊢ (0 + 2) = 2 |
36 | 15, 4, 12, 33, 35 | decaddi 12762 | . . . . 5 ⊢ ((;;;6700 · 6) + 2) = ;;;;40202 |
37 | 4cn 12322 | . . . . . 6 ⊢ 4 ∈ ℂ | |
38 | 6t4e24 12808 | . . . . . 6 ⊢ (6 · 4) = ;24 | |
39 | 30, 37, 38 | mulcomli 11248 | . . . . 5 ⊢ (4 · 6) = ;24 |
40 | 1, 6, 7, 19, 7, 12, 36, 39 | decmul1c 12767 | . . . 4 ⊢ (;;;;67004 · 6) = ;;;;;402024 |
41 | 30 | mullidi 11244 | . . . 4 ⊢ (1 · 6) = 6 |
42 | 1, 8, 9, 18, 40, 41 | decmul1 12766 | . . 3 ⊢ (;;;;;670041 · 6) = ;;;;;;4020246 |
43 | eqid 2725 | . . . 4 ⊢ ;;;;;402024 = ;;;;;402024 | |
44 | 4p1e5 12383 | . . . 4 ⊢ (4 + 1) = 5 | |
45 | 16, 7, 9, 43, 44 | decaddi 12762 | . . 3 ⊢ (;;;;;402024 + 1) = ;;;;;402025 |
46 | 17, 1, 7, 42, 45, 26 | decaddci2 12764 | . 2 ⊢ ((;;;;;670041 · 6) + 4) = ;;;;;;4020250 |
47 | 1, 10, 2, 11, 12, 7, 46, 28 | decmul1c 12767 | 1 ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7413 0cc0 11133 1c1 11134 · cmul 11138 2c2 12292 3c3 12293 4c4 12294 5c5 12295 6c6 12296 7c7 12297 ;cdc 12702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-ltxr 11278 df-sub 11471 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-dec 12703 |
This theorem is referenced by: fmtno5fac 46981 |
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