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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5 41979. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem1 | ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 11505 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 5nn0 11504 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
3 | 1, 2 | deccl 11704 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 2 | deccl 11704 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 11502 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 11704 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2760 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 9nn0 11508 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
9 | 5, 8 | deccl 11704 | . . . . 5 ⊢ ;39 ∈ ℕ0 |
10 | 9, 5 | deccl 11704 | . . . 4 ⊢ ;;393 ∈ ℕ0 |
11 | 1nn0 11500 | . . . 4 ⊢ 1 ∈ ℕ0 | |
12 | 10, 11 | deccl 11704 | . . 3 ⊢ ;;;3931 ∈ ℕ0 |
13 | 8nn0 11507 | . . 3 ⊢ 8 ∈ ℕ0 | |
14 | eqid 2760 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 0nn0 11499 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | 0p1e1 11324 | . . . . 5 ⊢ (0 + 1) = 1 | |
17 | eqid 2760 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2760 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t6e36 11838 | . . . . . . . . 9 ⊢ (6 · 6) = ;36 | |
20 | 6p3e9 11362 | . . . . . . . . 9 ⊢ (6 + 3) = 9 | |
21 | 5, 1, 5, 19, 20 | decaddi 11771 | . . . . . . . 8 ⊢ ((6 · 6) + 3) = ;39 |
22 | 6cn 11294 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
23 | 5cn 11292 | . . . . . . . . 9 ⊢ 5 ∈ ℂ | |
24 | 6t5e30 11836 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
25 | 22, 23, 24 | mulcomli 10239 | . . . . . . . 8 ⊢ (5 · 6) = ;30 |
26 | 1, 1, 2, 18, 15, 5, 21, 25 | decmul1c 11779 | . . . . . . 7 ⊢ (;65 · 6) = ;;390 |
27 | 3cn 11287 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
28 | 27 | addid2i 10416 | . . . . . . 7 ⊢ (0 + 3) = 3 |
29 | 9, 15, 5, 26, 28 | decaddi 11771 | . . . . . 6 ⊢ ((;65 · 6) + 3) = ;;393 |
30 | 1, 3, 2, 17, 15, 5, 29, 25 | decmul1c 11779 | . . . . 5 ⊢ (;;655 · 6) = ;;;3930 |
31 | 10, 15, 16, 30 | decsuc 11727 | . . . 4 ⊢ ((;;655 · 6) + 1) = ;;;3931 |
32 | 6t3e18 11834 | . . . . 5 ⊢ (6 · 3) = ;18 | |
33 | 22, 27, 32 | mulcomli 10239 | . . . 4 ⊢ (3 · 6) = ;18 |
34 | 1, 4, 5, 14, 13, 11, 31, 33 | decmul1c 11779 | . . 3 ⊢ (;;;6553 · 6) = ;;;;39318 |
35 | 1p1e2 11326 | . . . 4 ⊢ (1 + 1) = 2 | |
36 | eqid 2760 | . . . 4 ⊢ ;;;3931 = ;;;3931 | |
37 | 10, 11, 35, 36 | decsuc 11727 | . . 3 ⊢ (;;;3931 + 1) = ;;;3932 |
38 | 8p3e11 11804 | . . 3 ⊢ (8 + 3) = ;11 | |
39 | 12, 13, 5, 34, 37, 11, 38 | decaddci 11772 | . 2 ⊢ ((;;;6553 · 6) + 3) = ;;;;39321 |
40 | 1, 6, 1, 7, 1, 5, 39, 19 | decmul1c 11779 | 1 ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 (class class class)co 6813 0cc0 10128 1c1 10129 · cmul 10133 2c2 11262 3c3 11263 5c5 11265 6c6 11266 8c8 11268 9c9 11269 ;cdc 11685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-ltxr 10271 df-sub 10460 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-dec 11686 |
This theorem is referenced by: fmtno5lem4 41978 |
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