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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fmtno5 42504. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem3 | ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 11667 | . 2 ⊢ 3 ∈ ℕ0 | |
2 | 6nn0 11670 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 5nn0 11669 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
4 | 2, 3 | deccl 11865 | . . . 4 ⊢ ;65 ∈ ℕ0 |
5 | 4, 3 | deccl 11865 | . . 3 ⊢ ;;655 ∈ ℕ0 |
6 | 5, 1 | deccl 11865 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2778 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 8nn0 11672 | . 2 ⊢ 8 ∈ ℕ0 | |
9 | 1nn0 11665 | . 2 ⊢ 1 ∈ ℕ0 | |
10 | 9nn0 11673 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
11 | 9, 10 | deccl 11865 | . . . . 5 ⊢ ;19 ∈ ℕ0 |
12 | 11, 2 | deccl 11865 | . . . 4 ⊢ ;;196 ∈ ℕ0 |
13 | 12, 3 | deccl 11865 | . . 3 ⊢ ;;;1965 ∈ ℕ0 |
14 | 5p1e6 11534 | . . . 4 ⊢ (5 + 1) = 6 | |
15 | eqid 2778 | . . . 4 ⊢ ;;;1965 = ;;;1965 | |
16 | 12, 3, 14, 15 | decsuc 11882 | . . 3 ⊢ (;;;1965 + 1) = ;;;1966 |
17 | eqid 2778 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
18 | eqid 2778 | . . . . 5 ⊢ ;;655 = ;;655 | |
19 | eqid 2778 | . . . . . . 7 ⊢ ;65 = ;65 | |
20 | 8p1e9 11537 | . . . . . . . 8 ⊢ (8 + 1) = 9 | |
21 | 6t3e18 11957 | . . . . . . . 8 ⊢ (6 · 3) = ;18 | |
22 | 9, 8, 20, 21 | decsuc 11882 | . . . . . . 7 ⊢ ((6 · 3) + 1) = ;19 |
23 | 5t3e15 11953 | . . . . . . 7 ⊢ (5 · 3) = ;15 | |
24 | 1, 2, 3, 19, 3, 9, 22, 23 | decmul1c 11917 | . . . . . 6 ⊢ (;65 · 3) = ;;195 |
25 | 11, 3, 14, 24 | decsuc 11882 | . . . . 5 ⊢ ((;65 · 3) + 1) = ;;196 |
26 | 1, 4, 3, 18, 3, 9, 25, 23 | decmul1c 11917 | . . . 4 ⊢ (;;655 · 3) = ;;;1965 |
27 | 3t3e9 11554 | . . . 4 ⊢ (3 · 3) = 9 | |
28 | 1, 5, 1, 17, 26, 27 | decmul1 11915 | . . 3 ⊢ (;;;6553 · 3) = ;;;;19659 |
29 | 13, 16, 28 | decsucc 11892 | . 2 ⊢ ((;;;6553 · 3) + 1) = ;;;;19660 |
30 | 1, 6, 2, 7, 8, 9, 29, 21 | decmul1c 11917 | 1 ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 (class class class)co 6924 0cc0 10274 1c1 10275 · cmul 10279 3c3 11436 5c5 11438 6c6 11439 8c8 11441 9c9 11442 ;cdc 11850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-ltxr 10418 df-sub 10610 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-dec 11851 |
This theorem is referenced by: fmtno5lem4 42503 |
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