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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 11246 | . . . . . . 7 ⊢ 0 ∈ V | |
2 | 1 | snid 4669 | . . . . . 6 ⊢ 0 ∈ {0} |
3 | fzo01 13754 | . . . . . 6 ⊢ (0..^1) = {0} | |
4 | 2, 3 | eleqtrri 2828 | . . . . 5 ⊢ 0 ∈ (0..^1) |
5 | 2cn 12325 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
6 | exp0 14070 | . . . . . . 7 ⊢ (2 ∈ ℂ → (2↑0) = 1) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ (2↑0) = 1 |
8 | 7 | oveq2i 7437 | . . . . 5 ⊢ (0..^(2↑0)) = (0..^1) |
9 | 4, 8 | eleqtrri 2828 | . . . 4 ⊢ 0 ∈ (0..^(2↑0)) |
10 | 0z 12607 | . . . . 5 ⊢ 0 ∈ ℤ | |
11 | 0nn0 12525 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | bitsfzo 16417 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℕ0) → (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0))) | |
13 | 10, 11, 12 | mp2an 690 | . . . 4 ⊢ (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0)) |
14 | 9, 13 | mpbi 229 | . . 3 ⊢ (bits‘0) ⊆ (0..^0) |
15 | fzo0 13696 | . . 3 ⊢ (0..^0) = ∅ | |
16 | 14, 15 | sseqtri 4018 | . 2 ⊢ (bits‘0) ⊆ ∅ |
17 | 0ss 4400 | . 2 ⊢ ∅ ⊆ (bits‘0) | |
18 | 16, 17 | eqssi 3998 | 1 ⊢ (bits‘0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 ∅c0 4326 {csn 4632 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 0cc0 11146 1c1 11147 2c2 12305 ℕ0cn0 12510 ℤcz 12596 ..^cfzo 13667 ↑cexp 14066 bitscbits 16401 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-dvds 16239 df-bits 16404 |
This theorem is referenced by: m1bits 16422 sadcadd 16440 sadadd2 16442 bitsres 16455 smumullem 16474 eulerpartgbij 34025 eulerpartlemmf 34028 eulerpartlemgvv 34029 eulerpartlemgh 34031 |
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