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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5 45835. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem2 | ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 12438 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 6nn0 12439 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 2, 1 | deccl 12638 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 1 | deccl 12638 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 12436 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 12638 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2733 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 0nn0 12433 | . 2 ⊢ 0 ∈ ℕ0 | |
9 | 2nn0 12435 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 5, 9 | deccl 12638 | . . . . 5 ⊢ ;32 ∈ ℕ0 |
11 | 7nn0 12440 | . . . . 5 ⊢ 7 ∈ ℕ0 | |
12 | 10, 11 | deccl 12638 | . . . 4 ⊢ ;;327 ∈ ℕ0 |
13 | 12, 2 | deccl 12638 | . . 3 ⊢ ;;;3276 ∈ ℕ0 |
14 | eqid 2733 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 1nn0 12434 | . . . 4 ⊢ 1 ∈ ℕ0 | |
16 | 5p1e6 12305 | . . . . 5 ⊢ (5 + 1) = 6 | |
17 | eqid 2733 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2733 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t5e30 12730 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
20 | 2cn 12233 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
21 | 20 | addid2i 11348 | . . . . . . . . 9 ⊢ (0 + 2) = 2 |
22 | 5, 8, 9, 19, 21 | decaddi 12683 | . . . . . . . 8 ⊢ ((6 · 5) + 2) = ;32 |
23 | 5t5e25 12726 | . . . . . . . 8 ⊢ (5 · 5) = ;25 | |
24 | 1, 2, 1, 18, 1, 9, 22, 23 | decmul1c 12688 | . . . . . . 7 ⊢ (;65 · 5) = ;;325 |
25 | 5p2e7 12314 | . . . . . . 7 ⊢ (5 + 2) = 7 | |
26 | 10, 1, 9, 24, 25 | decaddi 12683 | . . . . . 6 ⊢ ((;65 · 5) + 2) = ;;327 |
27 | 1, 3, 1, 17, 1, 9, 26, 23 | decmul1c 12688 | . . . . 5 ⊢ (;;655 · 5) = ;;;3275 |
28 | 12, 1, 16, 27 | decsuc 12654 | . . . 4 ⊢ ((;;655 · 5) + 1) = ;;;3276 |
29 | 5cn 12246 | . . . . 5 ⊢ 5 ∈ ℂ | |
30 | 3cn 12239 | . . . . 5 ⊢ 3 ∈ ℂ | |
31 | 5t3e15 12724 | . . . . 5 ⊢ (5 · 3) = ;15 | |
32 | 29, 30, 31 | mulcomli 11169 | . . . 4 ⊢ (3 · 5) = ;15 |
33 | 1, 4, 5, 14, 1, 15, 28, 32 | decmul1c 12688 | . . 3 ⊢ (;;;6553 · 5) = ;;;;32765 |
34 | 5p3e8 12315 | . . 3 ⊢ (5 + 3) = 8 | |
35 | 13, 1, 5, 33, 34 | decaddi 12683 | . 2 ⊢ ((;;;6553 · 5) + 3) = ;;;;32768 |
36 | 1, 6, 2, 7, 8, 5, 35, 19 | decmul1c 12688 | 1 ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7358 0cc0 11056 1c1 11057 · cmul 11061 2c2 12213 3c3 12214 5c5 12216 6c6 12217 7c7 12218 8c8 12219 ;cdc 12623 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 df-sub 11392 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-dec 12624 |
This theorem is referenced by: fmtno5lem4 45834 |
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