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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5fac 44099. (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 2798 | . 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 2798 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
20 | eqid 2798 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
21 | eqid 2798 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
22 | eqid 2798 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
23 | eqid 2798 | . . . . . . . . 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 10818 | . . . . . . . 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 10635 | . . . 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 1538 (class class class)co 7135 0cc0 10526 1c1 10527 · cmul 10531 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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-nn 11626 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 44099 |
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