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Mirrors > Home > MPE Home > Th. List > 0bits | Structured version Visualization version GIF version |
Description: The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.) |
Ref | Expression |
---|---|
0bits | ⊢ (bits‘0) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10034 | . . . . . . 7 ⊢ 0 ∈ V | |
2 | 1 | snid 4208 | . . . . . 6 ⊢ 0 ∈ {0} |
3 | fzo01 12550 | . . . . . 6 ⊢ (0..^1) = {0} | |
4 | 2, 3 | eleqtrri 2700 | . . . . 5 ⊢ 0 ∈ (0..^1) |
5 | 2cn 11091 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
6 | exp0 12864 | . . . . . . 7 ⊢ (2 ∈ ℂ → (2↑0) = 1) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ (2↑0) = 1 |
8 | 7 | oveq2i 6661 | . . . . 5 ⊢ (0..^(2↑0)) = (0..^1) |
9 | 4, 8 | eleqtrri 2700 | . . . 4 ⊢ 0 ∈ (0..^(2↑0)) |
10 | 0z 11388 | . . . . 5 ⊢ 0 ∈ ℤ | |
11 | 0nn0 11307 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | bitsfzo 15157 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℕ0) → (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0))) | |
13 | 10, 11, 12 | mp2an 708 | . . . 4 ⊢ (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0)) |
14 | 9, 13 | mpbi 220 | . . 3 ⊢ (bits‘0) ⊆ (0..^0) |
15 | fzo0 12492 | . . 3 ⊢ (0..^0) = ∅ | |
16 | 14, 15 | sseqtri 3637 | . 2 ⊢ (bits‘0) ⊆ ∅ |
17 | 0ss 3972 | . 2 ⊢ ∅ ⊆ (bits‘0) | |
18 | 16, 17 | eqssi 3619 | 1 ⊢ (bits‘0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 {csn 4177 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 2c2 11070 ℕ0cn0 11292 ℤcz 11377 ..^cfzo 12465 ↑cexp 12860 bitscbits 15141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-dvds 14984 df-bits 15144 |
This theorem is referenced by: m1bits 15162 sadcadd 15180 sadadd2 15182 bitsres 15195 smumullem 15214 eulerpartgbij 30434 eulerpartlemmf 30437 eulerpartlemgvv 30438 eulerpartlemgh 30440 |
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