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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5 46802. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem2 | ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 12496 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 6nn0 12497 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 2, 1 | deccl 12696 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 1 | deccl 12696 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 12494 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 12696 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2726 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 0nn0 12491 | . 2 ⊢ 0 ∈ ℕ0 | |
9 | 2nn0 12493 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 5, 9 | deccl 12696 | . . . . 5 ⊢ ;32 ∈ ℕ0 |
11 | 7nn0 12498 | . . . . 5 ⊢ 7 ∈ ℕ0 | |
12 | 10, 11 | deccl 12696 | . . . 4 ⊢ ;;327 ∈ ℕ0 |
13 | 12, 2 | deccl 12696 | . . 3 ⊢ ;;;3276 ∈ ℕ0 |
14 | eqid 2726 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 1nn0 12492 | . . . 4 ⊢ 1 ∈ ℕ0 | |
16 | 5p1e6 12363 | . . . . 5 ⊢ (5 + 1) = 6 | |
17 | eqid 2726 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2726 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t5e30 12788 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
20 | 2cn 12291 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
21 | 20 | addlidi 11406 | . . . . . . . . 9 ⊢ (0 + 2) = 2 |
22 | 5, 8, 9, 19, 21 | decaddi 12741 | . . . . . . . 8 ⊢ ((6 · 5) + 2) = ;32 |
23 | 5t5e25 12784 | . . . . . . . 8 ⊢ (5 · 5) = ;25 | |
24 | 1, 2, 1, 18, 1, 9, 22, 23 | decmul1c 12746 | . . . . . . 7 ⊢ (;65 · 5) = ;;325 |
25 | 5p2e7 12372 | . . . . . . 7 ⊢ (5 + 2) = 7 | |
26 | 10, 1, 9, 24, 25 | decaddi 12741 | . . . . . 6 ⊢ ((;65 · 5) + 2) = ;;327 |
27 | 1, 3, 1, 17, 1, 9, 26, 23 | decmul1c 12746 | . . . . 5 ⊢ (;;655 · 5) = ;;;3275 |
28 | 12, 1, 16, 27 | decsuc 12712 | . . . 4 ⊢ ((;;655 · 5) + 1) = ;;;3276 |
29 | 5cn 12304 | . . . . 5 ⊢ 5 ∈ ℂ | |
30 | 3cn 12297 | . . . . 5 ⊢ 3 ∈ ℂ | |
31 | 5t3e15 12782 | . . . . 5 ⊢ (5 · 3) = ;15 | |
32 | 29, 30, 31 | mulcomli 11227 | . . . 4 ⊢ (3 · 5) = ;15 |
33 | 1, 4, 5, 14, 1, 15, 28, 32 | decmul1c 12746 | . . 3 ⊢ (;;;6553 · 5) = ;;;;32765 |
34 | 5p3e8 12373 | . . 3 ⊢ (5 + 3) = 8 | |
35 | 13, 1, 5, 33, 34 | decaddi 12741 | . 2 ⊢ ((;;;6553 · 5) + 3) = ;;;;32768 |
36 | 1, 6, 2, 7, 8, 5, 35, 19 | decmul1c 12746 | 1 ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7405 0cc0 11112 1c1 11113 · cmul 11117 2c2 12271 3c3 12272 5c5 12274 6c6 12275 7c7 12276 8c8 12277 ;cdc 12681 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-sub 11450 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-dec 12682 |
This theorem is referenced by: fmtno5lem4 46801 |
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