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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fmtno5 47034. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem3 | ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 12523 | . 2 ⊢ 3 ∈ ℕ0 | |
2 | 6nn0 12526 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 5nn0 12525 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
4 | 2, 3 | deccl 12725 | . . . 4 ⊢ ;65 ∈ ℕ0 |
5 | 4, 3 | deccl 12725 | . . 3 ⊢ ;;655 ∈ ℕ0 |
6 | 5, 1 | deccl 12725 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2725 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 8nn0 12528 | . 2 ⊢ 8 ∈ ℕ0 | |
9 | 1nn0 12521 | . 2 ⊢ 1 ∈ ℕ0 | |
10 | 9nn0 12529 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
11 | 9, 10 | deccl 12725 | . . . . 5 ⊢ ;19 ∈ ℕ0 |
12 | 11, 2 | deccl 12725 | . . . 4 ⊢ ;;196 ∈ ℕ0 |
13 | 12, 3 | deccl 12725 | . . 3 ⊢ ;;;1965 ∈ ℕ0 |
14 | 5p1e6 12392 | . . . 4 ⊢ (5 + 1) = 6 | |
15 | eqid 2725 | . . . 4 ⊢ ;;;1965 = ;;;1965 | |
16 | 12, 3, 14, 15 | decsuc 12741 | . . 3 ⊢ (;;;1965 + 1) = ;;;1966 |
17 | eqid 2725 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
18 | eqid 2725 | . . . . 5 ⊢ ;;655 = ;;655 | |
19 | eqid 2725 | . . . . . . 7 ⊢ ;65 = ;65 | |
20 | 8p1e9 12395 | . . . . . . . 8 ⊢ (8 + 1) = 9 | |
21 | 6t3e18 12815 | . . . . . . . 8 ⊢ (6 · 3) = ;18 | |
22 | 9, 8, 20, 21 | decsuc 12741 | . . . . . . 7 ⊢ ((6 · 3) + 1) = ;19 |
23 | 5t3e15 12811 | . . . . . . 7 ⊢ (5 · 3) = ;15 | |
24 | 1, 2, 3, 19, 3, 9, 22, 23 | decmul1c 12775 | . . . . . 6 ⊢ (;65 · 3) = ;;195 |
25 | 11, 3, 14, 24 | decsuc 12741 | . . . . 5 ⊢ ((;65 · 3) + 1) = ;;196 |
26 | 1, 4, 3, 18, 3, 9, 25, 23 | decmul1c 12775 | . . . 4 ⊢ (;;655 · 3) = ;;;1965 |
27 | 3t3e9 12412 | . . . 4 ⊢ (3 · 3) = 9 | |
28 | 1, 5, 1, 17, 26, 27 | decmul1 12774 | . . 3 ⊢ (;;;6553 · 3) = ;;;;19659 |
29 | 13, 16, 28 | decsucc 12751 | . 2 ⊢ ((;;;6553 · 3) + 1) = ;;;;19660 |
30 | 1, 6, 2, 7, 8, 9, 29, 21 | decmul1c 12775 | 1 ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7419 0cc0 11140 1c1 11141 · cmul 11145 3c3 12301 5c5 12303 6c6 12304 8c8 12306 9c9 12307 ;cdc 12710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-sub 11478 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-dec 12711 |
This theorem is referenced by: fmtno5lem4 47033 |
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