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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fmtno5 44727. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem3 | ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 12137 | . 2 ⊢ 3 ∈ ℕ0 | |
2 | 6nn0 12140 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 5nn0 12139 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
4 | 2, 3 | deccl 12337 | . . . 4 ⊢ ;65 ∈ ℕ0 |
5 | 4, 3 | deccl 12337 | . . 3 ⊢ ;;655 ∈ ℕ0 |
6 | 5, 1 | deccl 12337 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2739 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 8nn0 12142 | . 2 ⊢ 8 ∈ ℕ0 | |
9 | 1nn0 12135 | . 2 ⊢ 1 ∈ ℕ0 | |
10 | 9nn0 12143 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
11 | 9, 10 | deccl 12337 | . . . . 5 ⊢ ;19 ∈ ℕ0 |
12 | 11, 2 | deccl 12337 | . . . 4 ⊢ ;;196 ∈ ℕ0 |
13 | 12, 3 | deccl 12337 | . . 3 ⊢ ;;;1965 ∈ ℕ0 |
14 | 5p1e6 12006 | . . . 4 ⊢ (5 + 1) = 6 | |
15 | eqid 2739 | . . . 4 ⊢ ;;;1965 = ;;;1965 | |
16 | 12, 3, 14, 15 | decsuc 12353 | . . 3 ⊢ (;;;1965 + 1) = ;;;1966 |
17 | eqid 2739 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
18 | eqid 2739 | . . . . 5 ⊢ ;;655 = ;;655 | |
19 | eqid 2739 | . . . . . . 7 ⊢ ;65 = ;65 | |
20 | 8p1e9 12009 | . . . . . . . 8 ⊢ (8 + 1) = 9 | |
21 | 6t3e18 12427 | . . . . . . . 8 ⊢ (6 · 3) = ;18 | |
22 | 9, 8, 20, 21 | decsuc 12353 | . . . . . . 7 ⊢ ((6 · 3) + 1) = ;19 |
23 | 5t3e15 12423 | . . . . . . 7 ⊢ (5 · 3) = ;15 | |
24 | 1, 2, 3, 19, 3, 9, 22, 23 | decmul1c 12387 | . . . . . 6 ⊢ (;65 · 3) = ;;195 |
25 | 11, 3, 14, 24 | decsuc 12353 | . . . . 5 ⊢ ((;65 · 3) + 1) = ;;196 |
26 | 1, 4, 3, 18, 3, 9, 25, 23 | decmul1c 12387 | . . . 4 ⊢ (;;655 · 3) = ;;;1965 |
27 | 3t3e9 12026 | . . . 4 ⊢ (3 · 3) = 9 | |
28 | 1, 5, 1, 17, 26, 27 | decmul1 12386 | . . 3 ⊢ (;;;6553 · 3) = ;;;;19659 |
29 | 13, 16, 28 | decsucc 12363 | . 2 ⊢ ((;;;6553 · 3) + 1) = ;;;;19660 |
30 | 1, 6, 2, 7, 8, 9, 29, 21 | decmul1c 12387 | 1 ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 (class class class)co 7234 0cc0 10758 1c1 10759 · cmul 10763 3c3 11915 5c5 11917 6c6 11918 8c8 11920 9c9 11921 ;cdc 12322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-pnf 10898 df-mnf 10899 df-ltxr 10901 df-sub 11093 df-nn 11860 df-2 11922 df-3 11923 df-4 11924 df-5 11925 df-6 11926 df-7 11927 df-8 11928 df-9 11929 df-n0 12120 df-dec 12323 |
This theorem is referenced by: fmtno5lem4 44726 |
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