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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 11253 | . . . . . . 7 ⊢ 0 ∈ V | |
2 | 1 | snid 4667 | . . . . . 6 ⊢ 0 ∈ {0} |
3 | fzo01 13783 | . . . . . 6 ⊢ (0..^1) = {0} | |
4 | 2, 3 | eleqtrri 2838 | . . . . 5 ⊢ 0 ∈ (0..^1) |
5 | 2cn 12339 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
6 | exp0 14103 | . . . . . . 7 ⊢ (2 ∈ ℂ → (2↑0) = 1) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ (2↑0) = 1 |
8 | 7 | oveq2i 7442 | . . . . 5 ⊢ (0..^(2↑0)) = (0..^1) |
9 | 4, 8 | eleqtrri 2838 | . . . 4 ⊢ 0 ∈ (0..^(2↑0)) |
10 | 0z 12622 | . . . . 5 ⊢ 0 ∈ ℤ | |
11 | 0nn0 12539 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | bitsfzo 16469 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℕ0) → (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0))) | |
13 | 10, 11, 12 | mp2an 692 | . . . 4 ⊢ (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0)) |
14 | 9, 13 | mpbi 230 | . . 3 ⊢ (bits‘0) ⊆ (0..^0) |
15 | fzo0 13720 | . . 3 ⊢ (0..^0) = ∅ | |
16 | 14, 15 | sseqtri 4032 | . 2 ⊢ (bits‘0) ⊆ ∅ |
17 | 0ss 4406 | . 2 ⊢ ∅ ⊆ (bits‘0) | |
18 | 16, 17 | eqssi 4012 | 1 ⊢ (bits‘0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 {csn 4631 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ..^cfzo 13691 ↑cexp 14099 bitscbits 16453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-dvds 16288 df-bits 16456 |
This theorem is referenced by: m1bits 16474 sadcadd 16492 sadadd2 16494 bitsres 16507 smumullem 16526 eulerpartgbij 34354 eulerpartlemmf 34357 eulerpartlemgvv 34358 eulerpartlemgh 34360 |
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