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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5 45823. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem1 | ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 12441 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 5nn0 12440 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
3 | 1, 2 | deccl 12640 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 2 | deccl 12640 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 12438 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 12640 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2737 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 9nn0 12444 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
9 | 5, 8 | deccl 12640 | . . . . 5 ⊢ ;39 ∈ ℕ0 |
10 | 9, 5 | deccl 12640 | . . . 4 ⊢ ;;393 ∈ ℕ0 |
11 | 1nn0 12436 | . . . 4 ⊢ 1 ∈ ℕ0 | |
12 | 10, 11 | deccl 12640 | . . 3 ⊢ ;;;3931 ∈ ℕ0 |
13 | 8nn0 12443 | . . 3 ⊢ 8 ∈ ℕ0 | |
14 | eqid 2737 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 0nn0 12435 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | 0p1e1 12282 | . . . . 5 ⊢ (0 + 1) = 1 | |
17 | eqid 2737 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2737 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t6e36 12733 | . . . . . . . . 9 ⊢ (6 · 6) = ;36 | |
20 | 6p3e9 12320 | . . . . . . . . 9 ⊢ (6 + 3) = 9 | |
21 | 5, 1, 5, 19, 20 | decaddi 12685 | . . . . . . . 8 ⊢ ((6 · 6) + 3) = ;39 |
22 | 6cn 12251 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
23 | 5cn 12248 | . . . . . . . . 9 ⊢ 5 ∈ ℂ | |
24 | 6t5e30 12732 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
25 | 22, 23, 24 | mulcomli 11171 | . . . . . . . 8 ⊢ (5 · 6) = ;30 |
26 | 1, 1, 2, 18, 15, 5, 21, 25 | decmul1c 12690 | . . . . . . 7 ⊢ (;65 · 6) = ;;390 |
27 | 3cn 12241 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
28 | 27 | addid2i 11350 | . . . . . . 7 ⊢ (0 + 3) = 3 |
29 | 9, 15, 5, 26, 28 | decaddi 12685 | . . . . . 6 ⊢ ((;65 · 6) + 3) = ;;393 |
30 | 1, 3, 2, 17, 15, 5, 29, 25 | decmul1c 12690 | . . . . 5 ⊢ (;;655 · 6) = ;;;3930 |
31 | 10, 15, 16, 30 | decsuc 12656 | . . . 4 ⊢ ((;;655 · 6) + 1) = ;;;3931 |
32 | 6t3e18 12730 | . . . . 5 ⊢ (6 · 3) = ;18 | |
33 | 22, 27, 32 | mulcomli 11171 | . . . 4 ⊢ (3 · 6) = ;18 |
34 | 1, 4, 5, 14, 13, 11, 31, 33 | decmul1c 12690 | . . 3 ⊢ (;;;6553 · 6) = ;;;;39318 |
35 | 1p1e2 12285 | . . . 4 ⊢ (1 + 1) = 2 | |
36 | eqid 2737 | . . . 4 ⊢ ;;;3931 = ;;;3931 | |
37 | 10, 11, 35, 36 | decsuc 12656 | . . 3 ⊢ (;;;3931 + 1) = ;;;3932 |
38 | 8p3e11 12706 | . . 3 ⊢ (8 + 3) = ;11 | |
39 | 12, 13, 5, 34, 37, 11, 38 | decaddci 12686 | . 2 ⊢ ((;;;6553 · 6) + 3) = ;;;;39321 |
40 | 1, 6, 1, 7, 1, 5, 39, 19 | decmul1c 12690 | 1 ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7362 0cc0 11058 1c1 11059 · cmul 11063 2c2 12215 3c3 12216 5c5 12218 6c6 12219 8c8 12221 9c9 12222 ;cdc 12625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-ltxr 11201 df-sub 11394 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-dec 12626 |
This theorem is referenced by: fmtno5lem4 45822 |
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