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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5fac 43621. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5faclem1 | ⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 11904 | . 2 ⊢ 4 ∈ ℕ0 | |
2 | 6nn0 11906 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
3 | 7nn0 11907 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
4 | 2, 3 | deccl 12101 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
5 | 0nn0 11900 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 4, 5 | deccl 12101 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
7 | 6, 5 | deccl 12101 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
8 | 7, 1 | deccl 12101 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 11901 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12101 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | eqid 2818 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
12 | 8nn0 11908 | . 2 ⊢ 8 ∈ ℕ0 | |
13 | 2nn0 11902 | . 2 ⊢ 2 ∈ ℕ0 | |
14 | 13, 2 | deccl 12101 | . . . . . . 7 ⊢ ;26 ∈ ℕ0 |
15 | 14, 12 | deccl 12101 | . . . . . 6 ⊢ ;;268 ∈ ℕ0 |
16 | 15, 5 | deccl 12101 | . . . . 5 ⊢ ;;;2680 ∈ ℕ0 |
17 | 16, 9 | deccl 12101 | . . . 4 ⊢ ;;;;26801 ∈ ℕ0 |
18 | 17, 2 | deccl 12101 | . . 3 ⊢ ;;;;;268016 ∈ ℕ0 |
19 | eqid 2818 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
20 | eqid 2818 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
21 | eqid 2818 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
22 | eqid 2818 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
23 | eqid 2818 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
24 | 6t4e24 12192 | . . . . . . . . . 10 ⊢ (6 · 4) = ;24 | |
25 | 4p2e6 11778 | . . . . . . . . . 10 ⊢ (4 + 2) = 6 | |
26 | 13, 1, 13, 24, 25 | decaddi 12146 | . . . . . . . . 9 ⊢ ((6 · 4) + 2) = ;26 |
27 | 7t4e28 12197 | . . . . . . . . 9 ⊢ (7 · 4) = ;28 | |
28 | 1, 2, 3, 23, 12, 13, 26, 27 | decmul1c 12151 | . . . . . . . 8 ⊢ (;67 · 4) = ;;268 |
29 | 4cn 11710 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
30 | 29 | mul02i 10817 | . . . . . . . 8 ⊢ (0 · 4) = 0 |
31 | 1, 4, 5, 22, 28, 30 | decmul1 12150 | . . . . . . 7 ⊢ (;;670 · 4) = ;;;2680 |
32 | 1, 6, 5, 21, 31, 30 | decmul1 12150 | . . . . . 6 ⊢ (;;;6700 · 4) = ;;;;26800 |
33 | 0p1e1 11747 | . . . . . 6 ⊢ (0 + 1) = 1 | |
34 | 16, 5, 9, 32, 33 | decaddi 12146 | . . . . 5 ⊢ ((;;;6700 · 4) + 1) = ;;;;26801 |
35 | 4t4e16 12185 | . . . . 5 ⊢ (4 · 4) = ;16 | |
36 | 1, 7, 1, 20, 2, 9, 34, 35 | decmul1c 12151 | . . . 4 ⊢ (;;;;67004 · 4) = ;;;;;268016 |
37 | 29 | mulid2i 10634 | . . . 4 ⊢ (1 · 4) = 4 |
38 | 1, 8, 9, 19, 36, 37 | decmul1 12150 | . . 3 ⊢ (;;;;;670041 · 4) = ;;;;;;2680164 |
39 | 18, 1, 13, 38, 25 | decaddi 12146 | . 2 ⊢ ((;;;;;670041 · 4) + 2) = ;;;;;;2680166 |
40 | 1, 10, 3, 11, 12, 13, 39, 27 | decmul1c 12151 | 1 ⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 0cc0 10525 1c1 10526 · cmul 10530 2c2 11680 4c4 11682 6c6 11684 7c7 11685 8c8 11686 ;cdc 12086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 |
This theorem is referenced by: fmtno5fac 43621 |
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