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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fmtno5 45823. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem3 | ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 12438 | . 2 ⊢ 3 ∈ ℕ0 | |
2 | 6nn0 12441 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 5nn0 12440 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
4 | 2, 3 | deccl 12640 | . . . 4 ⊢ ;65 ∈ ℕ0 |
5 | 4, 3 | deccl 12640 | . . 3 ⊢ ;;655 ∈ ℕ0 |
6 | 5, 1 | deccl 12640 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2737 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 8nn0 12443 | . 2 ⊢ 8 ∈ ℕ0 | |
9 | 1nn0 12436 | . 2 ⊢ 1 ∈ ℕ0 | |
10 | 9nn0 12444 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
11 | 9, 10 | deccl 12640 | . . . . 5 ⊢ ;19 ∈ ℕ0 |
12 | 11, 2 | deccl 12640 | . . . 4 ⊢ ;;196 ∈ ℕ0 |
13 | 12, 3 | deccl 12640 | . . 3 ⊢ ;;;1965 ∈ ℕ0 |
14 | 5p1e6 12307 | . . . 4 ⊢ (5 + 1) = 6 | |
15 | eqid 2737 | . . . 4 ⊢ ;;;1965 = ;;;1965 | |
16 | 12, 3, 14, 15 | decsuc 12656 | . . 3 ⊢ (;;;1965 + 1) = ;;;1966 |
17 | eqid 2737 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
18 | eqid 2737 | . . . . 5 ⊢ ;;655 = ;;655 | |
19 | eqid 2737 | . . . . . . 7 ⊢ ;65 = ;65 | |
20 | 8p1e9 12310 | . . . . . . . 8 ⊢ (8 + 1) = 9 | |
21 | 6t3e18 12730 | . . . . . . . 8 ⊢ (6 · 3) = ;18 | |
22 | 9, 8, 20, 21 | decsuc 12656 | . . . . . . 7 ⊢ ((6 · 3) + 1) = ;19 |
23 | 5t3e15 12726 | . . . . . . 7 ⊢ (5 · 3) = ;15 | |
24 | 1, 2, 3, 19, 3, 9, 22, 23 | decmul1c 12690 | . . . . . 6 ⊢ (;65 · 3) = ;;195 |
25 | 11, 3, 14, 24 | decsuc 12656 | . . . . 5 ⊢ ((;65 · 3) + 1) = ;;196 |
26 | 1, 4, 3, 18, 3, 9, 25, 23 | decmul1c 12690 | . . . 4 ⊢ (;;655 · 3) = ;;;1965 |
27 | 3t3e9 12327 | . . . 4 ⊢ (3 · 3) = 9 | |
28 | 1, 5, 1, 17, 26, 27 | decmul1 12689 | . . 3 ⊢ (;;;6553 · 3) = ;;;;19659 |
29 | 13, 16, 28 | decsucc 12666 | . 2 ⊢ ((;;;6553 · 3) + 1) = ;;;;19660 |
30 | 1, 6, 2, 7, 8, 9, 29, 21 | decmul1c 12690 | 1 ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7362 0cc0 11058 1c1 11059 · cmul 11063 3c3 12216 5c5 12218 6c6 12219 8c8 12221 9c9 12222 ;cdc 12625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-ltxr 11201 df-sub 11394 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-dec 12626 |
This theorem is referenced by: fmtno5lem4 45822 |
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