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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5fac 47154. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5faclem1 | ⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 12543 | . 2 ⊢ 4 ∈ ℕ0 | |
2 | 6nn0 12545 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
3 | 7nn0 12546 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
4 | 2, 3 | deccl 12744 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
5 | 0nn0 12539 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 4, 5 | deccl 12744 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
7 | 6, 5 | deccl 12744 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
8 | 7, 1 | deccl 12744 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 12540 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12744 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | eqid 2726 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
12 | 8nn0 12547 | . 2 ⊢ 8 ∈ ℕ0 | |
13 | 2nn0 12541 | . 2 ⊢ 2 ∈ ℕ0 | |
14 | 13, 2 | deccl 12744 | . . . . . . 7 ⊢ ;26 ∈ ℕ0 |
15 | 14, 12 | deccl 12744 | . . . . . 6 ⊢ ;;268 ∈ ℕ0 |
16 | 15, 5 | deccl 12744 | . . . . 5 ⊢ ;;;2680 ∈ ℕ0 |
17 | 16, 9 | deccl 12744 | . . . 4 ⊢ ;;;;26801 ∈ ℕ0 |
18 | 17, 2 | deccl 12744 | . . 3 ⊢ ;;;;;268016 ∈ ℕ0 |
19 | eqid 2726 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
20 | eqid 2726 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
21 | eqid 2726 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
22 | eqid 2726 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
23 | eqid 2726 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
24 | 6t4e24 12835 | . . . . . . . . . 10 ⊢ (6 · 4) = ;24 | |
25 | 4p2e6 12417 | . . . . . . . . . 10 ⊢ (4 + 2) = 6 | |
26 | 13, 1, 13, 24, 25 | decaddi 12789 | . . . . . . . . 9 ⊢ ((6 · 4) + 2) = ;26 |
27 | 7t4e28 12840 | . . . . . . . . 9 ⊢ (7 · 4) = ;28 | |
28 | 1, 2, 3, 23, 12, 13, 26, 27 | decmul1c 12794 | . . . . . . . 8 ⊢ (;67 · 4) = ;;268 |
29 | 4cn 12349 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
30 | 29 | mul02i 11453 | . . . . . . . 8 ⊢ (0 · 4) = 0 |
31 | 1, 4, 5, 22, 28, 30 | decmul1 12793 | . . . . . . 7 ⊢ (;;670 · 4) = ;;;2680 |
32 | 1, 6, 5, 21, 31, 30 | decmul1 12793 | . . . . . 6 ⊢ (;;;6700 · 4) = ;;;;26800 |
33 | 0p1e1 12386 | . . . . . 6 ⊢ (0 + 1) = 1 | |
34 | 16, 5, 9, 32, 33 | decaddi 12789 | . . . . 5 ⊢ ((;;;6700 · 4) + 1) = ;;;;26801 |
35 | 4t4e16 12828 | . . . . 5 ⊢ (4 · 4) = ;16 | |
36 | 1, 7, 1, 20, 2, 9, 34, 35 | decmul1c 12794 | . . . 4 ⊢ (;;;;67004 · 4) = ;;;;;268016 |
37 | 29 | mullidi 11269 | . . . 4 ⊢ (1 · 4) = 4 |
38 | 1, 8, 9, 19, 36, 37 | decmul1 12793 | . . 3 ⊢ (;;;;;670041 · 4) = ;;;;;;2680164 |
39 | 18, 1, 13, 38, 25 | decaddi 12789 | . 2 ⊢ ((;;;;;670041 · 4) + 2) = ;;;;;;2680166 |
40 | 1, 10, 3, 11, 12, 13, 39, 27 | decmul1c 12794 | 1 ⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7424 0cc0 11158 1c1 11159 · cmul 11163 2c2 12319 4c4 12321 6c6 12323 7c7 12324 8c8 12325 ;cdc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12730 |
This theorem is referenced by: fmtno5fac 47154 |
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