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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5fac 43764. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5faclem1 | ⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 11917 | . 2 ⊢ 4 ∈ ℕ0 | |
2 | 6nn0 11919 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
3 | 7nn0 11920 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
4 | 2, 3 | deccl 12114 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
5 | 0nn0 11913 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 4, 5 | deccl 12114 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
7 | 6, 5 | deccl 12114 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
8 | 7, 1 | deccl 12114 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 11914 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12114 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | eqid 2821 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
12 | 8nn0 11921 | . 2 ⊢ 8 ∈ ℕ0 | |
13 | 2nn0 11915 | . 2 ⊢ 2 ∈ ℕ0 | |
14 | 13, 2 | deccl 12114 | . . . . . . 7 ⊢ ;26 ∈ ℕ0 |
15 | 14, 12 | deccl 12114 | . . . . . 6 ⊢ ;;268 ∈ ℕ0 |
16 | 15, 5 | deccl 12114 | . . . . 5 ⊢ ;;;2680 ∈ ℕ0 |
17 | 16, 9 | deccl 12114 | . . . 4 ⊢ ;;;;26801 ∈ ℕ0 |
18 | 17, 2 | deccl 12114 | . . 3 ⊢ ;;;;;268016 ∈ ℕ0 |
19 | eqid 2821 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
20 | eqid 2821 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
21 | eqid 2821 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
22 | eqid 2821 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
23 | eqid 2821 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
24 | 6t4e24 12205 | . . . . . . . . . 10 ⊢ (6 · 4) = ;24 | |
25 | 4p2e6 11791 | . . . . . . . . . 10 ⊢ (4 + 2) = 6 | |
26 | 13, 1, 13, 24, 25 | decaddi 12159 | . . . . . . . . 9 ⊢ ((6 · 4) + 2) = ;26 |
27 | 7t4e28 12210 | . . . . . . . . 9 ⊢ (7 · 4) = ;28 | |
28 | 1, 2, 3, 23, 12, 13, 26, 27 | decmul1c 12164 | . . . . . . . 8 ⊢ (;67 · 4) = ;;268 |
29 | 4cn 11723 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
30 | 29 | mul02i 10829 | . . . . . . . 8 ⊢ (0 · 4) = 0 |
31 | 1, 4, 5, 22, 28, 30 | decmul1 12163 | . . . . . . 7 ⊢ (;;670 · 4) = ;;;2680 |
32 | 1, 6, 5, 21, 31, 30 | decmul1 12163 | . . . . . 6 ⊢ (;;;6700 · 4) = ;;;;26800 |
33 | 0p1e1 11760 | . . . . . 6 ⊢ (0 + 1) = 1 | |
34 | 16, 5, 9, 32, 33 | decaddi 12159 | . . . . 5 ⊢ ((;;;6700 · 4) + 1) = ;;;;26801 |
35 | 4t4e16 12198 | . . . . 5 ⊢ (4 · 4) = ;16 | |
36 | 1, 7, 1, 20, 2, 9, 34, 35 | decmul1c 12164 | . . . 4 ⊢ (;;;;67004 · 4) = ;;;;;268016 |
37 | 29 | mulid2i 10646 | . . . 4 ⊢ (1 · 4) = 4 |
38 | 1, 8, 9, 19, 36, 37 | decmul1 12163 | . . 3 ⊢ (;;;;;670041 · 4) = ;;;;;;2680164 |
39 | 18, 1, 13, 38, 25 | decaddi 12159 | . 2 ⊢ ((;;;;;670041 · 4) + 2) = ;;;;;;2680166 |
40 | 1, 10, 3, 11, 12, 13, 39, 27 | decmul1c 12164 | 1 ⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7156 0cc0 10537 1c1 10538 · cmul 10542 2c2 11693 4c4 11695 6c6 11697 7c7 11698 8c8 11699 ;cdc 12099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 |
This theorem is referenced by: fmtno5fac 43764 |
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