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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fmtno5 41469. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem3 | ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 11310 | . 2 ⊢ 3 ∈ ℕ0 | |
2 | 6nn0 11313 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 5nn0 11312 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
4 | 2, 3 | deccl 11512 | . . . 4 ⊢ ;65 ∈ ℕ0 |
5 | 4, 3 | deccl 11512 | . . 3 ⊢ ;;655 ∈ ℕ0 |
6 | 5, 1 | deccl 11512 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2622 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 8nn0 11315 | . 2 ⊢ 8 ∈ ℕ0 | |
9 | 1nn0 11308 | . 2 ⊢ 1 ∈ ℕ0 | |
10 | 9nn0 11316 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
11 | 9, 10 | deccl 11512 | . . . . 5 ⊢ ;19 ∈ ℕ0 |
12 | 11, 2 | deccl 11512 | . . . 4 ⊢ ;;196 ∈ ℕ0 |
13 | 12, 3 | deccl 11512 | . . 3 ⊢ ;;;1965 ∈ ℕ0 |
14 | 5p1e6 11155 | . . . 4 ⊢ (5 + 1) = 6 | |
15 | eqid 2622 | . . . 4 ⊢ ;;;1965 = ;;;1965 | |
16 | 12, 3, 14, 15 | decsuc 11535 | . . 3 ⊢ (;;;1965 + 1) = ;;;1966 |
17 | eqid 2622 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
18 | eqid 2622 | . . . . 5 ⊢ ;;655 = ;;655 | |
19 | eqid 2622 | . . . . . . 7 ⊢ ;65 = ;65 | |
20 | 8p1e9 11158 | . . . . . . . 8 ⊢ (8 + 1) = 9 | |
21 | 6t3e18 11642 | . . . . . . . 8 ⊢ (6 · 3) = ;18 | |
22 | 9, 8, 20, 21 | decsuc 11535 | . . . . . . 7 ⊢ ((6 · 3) + 1) = ;19 |
23 | 5t3e15 11635 | . . . . . . 7 ⊢ (5 · 3) = ;15 | |
24 | 1, 2, 3, 19, 3, 9, 22, 23 | decmul1c 11587 | . . . . . 6 ⊢ (;65 · 3) = ;;195 |
25 | 11, 3, 14, 24 | decsuc 11535 | . . . . 5 ⊢ ((;65 · 3) + 1) = ;;196 |
26 | 1, 4, 3, 18, 3, 9, 25, 23 | decmul1c 11587 | . . . 4 ⊢ (;;655 · 3) = ;;;1965 |
27 | 3t3e9 11180 | . . . 4 ⊢ (3 · 3) = 9 | |
28 | 1, 5, 1, 17, 10, 26, 27 | decmul1 11585 | . . 3 ⊢ (;;;6553 · 3) = ;;;;19659 |
29 | 13, 16, 28 | decsucc 11550 | . 2 ⊢ ((;;;6553 · 3) + 1) = ;;;;19660 |
30 | 1, 6, 2, 7, 8, 9, 29, 21 | decmul1c 11587 | 1 ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 (class class class)co 6650 0cc0 9936 1c1 9937 · cmul 9941 3c3 11071 5c5 11073 6c6 11074 8c8 11076 9c9 11077 ;cdc 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494 |
This theorem is referenced by: fmtno5lem4 41468 |
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