![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5 42240. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem2 | ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 11599 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 6nn0 11600 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 2, 1 | deccl 11795 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 1 | deccl 11795 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 11597 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 11795 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2798 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 0nn0 11594 | . 2 ⊢ 0 ∈ ℕ0 | |
9 | 2nn0 11596 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 5, 9 | deccl 11795 | . . . . 5 ⊢ ;32 ∈ ℕ0 |
11 | 7nn0 11601 | . . . . 5 ⊢ 7 ∈ ℕ0 | |
12 | 10, 11 | deccl 11795 | . . . 4 ⊢ ;;327 ∈ ℕ0 |
13 | 12, 2 | deccl 11795 | . . 3 ⊢ ;;;3276 ∈ ℕ0 |
14 | eqid 2798 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 1nn0 11595 | . . . 4 ⊢ 1 ∈ ℕ0 | |
16 | 5p1e6 11464 | . . . . 5 ⊢ (5 + 1) = 6 | |
17 | eqid 2798 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2798 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t5e30 11889 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
20 | 2cn 11385 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
21 | 20 | addid2i 10513 | . . . . . . . . 9 ⊢ (0 + 2) = 2 |
22 | 5, 8, 9, 19, 21 | decaddi 11841 | . . . . . . . 8 ⊢ ((6 · 5) + 2) = ;32 |
23 | 5t5e25 11885 | . . . . . . . 8 ⊢ (5 · 5) = ;25 | |
24 | 1, 2, 1, 18, 1, 9, 22, 23 | decmul1c 11847 | . . . . . . 7 ⊢ (;65 · 5) = ;;325 |
25 | 5p2e7 11473 | . . . . . . 7 ⊢ (5 + 2) = 7 | |
26 | 10, 1, 9, 24, 25 | decaddi 11841 | . . . . . 6 ⊢ ((;65 · 5) + 2) = ;;327 |
27 | 1, 3, 1, 17, 1, 9, 26, 23 | decmul1c 11847 | . . . . 5 ⊢ (;;655 · 5) = ;;;3275 |
28 | 12, 1, 16, 27 | decsuc 11812 | . . . 4 ⊢ ((;;655 · 5) + 1) = ;;;3276 |
29 | 5cn 11400 | . . . . 5 ⊢ 5 ∈ ℂ | |
30 | 3cn 11391 | . . . . 5 ⊢ 3 ∈ ℂ | |
31 | 5t3e15 11883 | . . . . 5 ⊢ (5 · 3) = ;15 | |
32 | 29, 30, 31 | mulcomli 10337 | . . . 4 ⊢ (3 · 5) = ;15 |
33 | 1, 4, 5, 14, 1, 15, 28, 32 | decmul1c 11847 | . . 3 ⊢ (;;;6553 · 5) = ;;;;32765 |
34 | 5p3e8 11474 | . . 3 ⊢ (5 + 3) = 8 | |
35 | 13, 1, 5, 33, 34 | decaddi 11841 | . 2 ⊢ ((;;;6553 · 5) + 3) = ;;;;32768 |
36 | 1, 6, 2, 7, 8, 5, 35, 19 | decmul1c 11847 | 1 ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 (class class class)co 6877 0cc0 10223 1c1 10224 · cmul 10228 2c2 11365 3c3 11366 5c5 11368 6c6 11369 7c7 11370 8c8 11371 ;cdc 11780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-pnf 10364 df-mnf 10365 df-ltxr 10367 df-sub 10557 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-dec 11781 |
This theorem is referenced by: fmtno5lem4 42239 |
Copyright terms: Public domain | W3C validator |