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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5 44591. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem2 | ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 12009 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 6nn0 12010 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 2, 1 | deccl 12207 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 1 | deccl 12207 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 12007 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 12207 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2739 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 0nn0 12004 | . 2 ⊢ 0 ∈ ℕ0 | |
9 | 2nn0 12006 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 5, 9 | deccl 12207 | . . . . 5 ⊢ ;32 ∈ ℕ0 |
11 | 7nn0 12011 | . . . . 5 ⊢ 7 ∈ ℕ0 | |
12 | 10, 11 | deccl 12207 | . . . 4 ⊢ ;;327 ∈ ℕ0 |
13 | 12, 2 | deccl 12207 | . . 3 ⊢ ;;;3276 ∈ ℕ0 |
14 | eqid 2739 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 1nn0 12005 | . . . 4 ⊢ 1 ∈ ℕ0 | |
16 | 5p1e6 11876 | . . . . 5 ⊢ (5 + 1) = 6 | |
17 | eqid 2739 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2739 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t5e30 12299 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
20 | 2cn 11804 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
21 | 20 | addid2i 10919 | . . . . . . . . 9 ⊢ (0 + 2) = 2 |
22 | 5, 8, 9, 19, 21 | decaddi 12252 | . . . . . . . 8 ⊢ ((6 · 5) + 2) = ;32 |
23 | 5t5e25 12295 | . . . . . . . 8 ⊢ (5 · 5) = ;25 | |
24 | 1, 2, 1, 18, 1, 9, 22, 23 | decmul1c 12257 | . . . . . . 7 ⊢ (;65 · 5) = ;;325 |
25 | 5p2e7 11885 | . . . . . . 7 ⊢ (5 + 2) = 7 | |
26 | 10, 1, 9, 24, 25 | decaddi 12252 | . . . . . 6 ⊢ ((;65 · 5) + 2) = ;;327 |
27 | 1, 3, 1, 17, 1, 9, 26, 23 | decmul1c 12257 | . . . . 5 ⊢ (;;655 · 5) = ;;;3275 |
28 | 12, 1, 16, 27 | decsuc 12223 | . . . 4 ⊢ ((;;655 · 5) + 1) = ;;;3276 |
29 | 5cn 11817 | . . . . 5 ⊢ 5 ∈ ℂ | |
30 | 3cn 11810 | . . . . 5 ⊢ 3 ∈ ℂ | |
31 | 5t3e15 12293 | . . . . 5 ⊢ (5 · 3) = ;15 | |
32 | 29, 30, 31 | mulcomli 10741 | . . . 4 ⊢ (3 · 5) = ;15 |
33 | 1, 4, 5, 14, 1, 15, 28, 32 | decmul1c 12257 | . . 3 ⊢ (;;;6553 · 5) = ;;;;32765 |
34 | 5p3e8 11886 | . . 3 ⊢ (5 + 3) = 8 | |
35 | 13, 1, 5, 33, 34 | decaddi 12252 | . 2 ⊢ ((;;;6553 · 5) + 3) = ;;;;32768 |
36 | 1, 6, 2, 7, 8, 5, 35, 19 | decmul1c 12257 | 1 ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7183 0cc0 10628 1c1 10629 · cmul 10633 2c2 11784 3c3 11785 5c5 11787 6c6 11788 7c7 11789 8c8 11790 ;cdc 12192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-ltxr 10771 df-sub 10963 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-dec 12193 |
This theorem is referenced by: fmtno5lem4 44590 |
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