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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5 46960. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem1 | ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 12523 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 5nn0 12522 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
3 | 1, 2 | deccl 12722 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 2 | deccl 12722 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 12520 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 12722 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2725 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 9nn0 12526 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
9 | 5, 8 | deccl 12722 | . . . . 5 ⊢ ;39 ∈ ℕ0 |
10 | 9, 5 | deccl 12722 | . . . 4 ⊢ ;;393 ∈ ℕ0 |
11 | 1nn0 12518 | . . . 4 ⊢ 1 ∈ ℕ0 | |
12 | 10, 11 | deccl 12722 | . . 3 ⊢ ;;;3931 ∈ ℕ0 |
13 | 8nn0 12525 | . . 3 ⊢ 8 ∈ ℕ0 | |
14 | eqid 2725 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 0nn0 12517 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | 0p1e1 12364 | . . . . 5 ⊢ (0 + 1) = 1 | |
17 | eqid 2725 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2725 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t6e36 12815 | . . . . . . . . 9 ⊢ (6 · 6) = ;36 | |
20 | 6p3e9 12402 | . . . . . . . . 9 ⊢ (6 + 3) = 9 | |
21 | 5, 1, 5, 19, 20 | decaddi 12767 | . . . . . . . 8 ⊢ ((6 · 6) + 3) = ;39 |
22 | 6cn 12333 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
23 | 5cn 12330 | . . . . . . . . 9 ⊢ 5 ∈ ℂ | |
24 | 6t5e30 12814 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
25 | 22, 23, 24 | mulcomli 11253 | . . . . . . . 8 ⊢ (5 · 6) = ;30 |
26 | 1, 1, 2, 18, 15, 5, 21, 25 | decmul1c 12772 | . . . . . . 7 ⊢ (;65 · 6) = ;;390 |
27 | 3cn 12323 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
28 | 27 | addlidi 11432 | . . . . . . 7 ⊢ (0 + 3) = 3 |
29 | 9, 15, 5, 26, 28 | decaddi 12767 | . . . . . 6 ⊢ ((;65 · 6) + 3) = ;;393 |
30 | 1, 3, 2, 17, 15, 5, 29, 25 | decmul1c 12772 | . . . . 5 ⊢ (;;655 · 6) = ;;;3930 |
31 | 10, 15, 16, 30 | decsuc 12738 | . . . 4 ⊢ ((;;655 · 6) + 1) = ;;;3931 |
32 | 6t3e18 12812 | . . . . 5 ⊢ (6 · 3) = ;18 | |
33 | 22, 27, 32 | mulcomli 11253 | . . . 4 ⊢ (3 · 6) = ;18 |
34 | 1, 4, 5, 14, 13, 11, 31, 33 | decmul1c 12772 | . . 3 ⊢ (;;;6553 · 6) = ;;;;39318 |
35 | 1p1e2 12367 | . . . 4 ⊢ (1 + 1) = 2 | |
36 | eqid 2725 | . . . 4 ⊢ ;;;3931 = ;;;3931 | |
37 | 10, 11, 35, 36 | decsuc 12738 | . . 3 ⊢ (;;;3931 + 1) = ;;;3932 |
38 | 8p3e11 12788 | . . 3 ⊢ (8 + 3) = ;11 | |
39 | 12, 13, 5, 34, 37, 11, 38 | decaddci 12768 | . 2 ⊢ ((;;;6553 · 6) + 3) = ;;;;39321 |
40 | 1, 6, 1, 7, 1, 5, 39, 19 | decmul1c 12772 | 1 ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7416 0cc0 11138 1c1 11139 · cmul 11143 2c2 12297 3c3 12298 5c5 12300 6c6 12301 8c8 12303 9c9 12304 ;cdc 12707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-sub 11476 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-dec 12708 |
This theorem is referenced by: fmtno5lem4 46959 |
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