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Mirrors > Home > MPE Home > Th. List > fzo0to42pr | Structured version Visualization version GIF version |
Description: A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.) |
Ref | Expression |
---|---|
fzo0to42pr | ⊢ (0..^4) = ({0, 1} ∪ {2, 3}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11186 | . . . 4 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 11188 | . . . 4 ⊢ 4 ∈ ℕ0 | |
3 | 2re 10967 | . . . . 5 ⊢ 2 ∈ ℝ | |
4 | 4re 10974 | . . . . 5 ⊢ 4 ∈ ℝ | |
5 | 2lt4 11075 | . . . . 5 ⊢ 2 < 4 | |
6 | 3, 4, 5 | ltleii 10039 | . . . 4 ⊢ 2 ≤ 4 |
7 | elfz2nn0 12300 | . . . 4 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ0 ∧ 4 ∈ ℕ0 ∧ 2 ≤ 4)) | |
8 | 1, 2, 6, 7 | mpbir3an 1237 | . . 3 ⊢ 2 ∈ (0...4) |
9 | fzosplit 12370 | . . 3 ⊢ (2 ∈ (0...4) → (0..^4) = ((0..^2) ∪ (2..^4))) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ (0..^4) = ((0..^2) ∪ (2..^4)) |
11 | fzo0to2pr 12420 | . . 3 ⊢ (0..^2) = {0, 1} | |
12 | 4z 11288 | . . . . 5 ⊢ 4 ∈ ℤ | |
13 | fzoval 12340 | . . . . 5 ⊢ (4 ∈ ℤ → (2..^4) = (2...(4 − 1))) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (2..^4) = (2...(4 − 1)) |
15 | 4cn 10975 | . . . . . . . 8 ⊢ 4 ∈ ℂ | |
16 | ax-1cn 9873 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
17 | 3cn 10972 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
18 | df-4 10958 | . . . . . . . . . 10 ⊢ 4 = (3 + 1) | |
19 | 17, 16 | addcomi 10106 | . . . . . . . . . 10 ⊢ (3 + 1) = (1 + 3) |
20 | 18, 19 | eqtri 2632 | . . . . . . . . 9 ⊢ 4 = (1 + 3) |
21 | 20 | eqcomi 2619 | . . . . . . . 8 ⊢ (1 + 3) = 4 |
22 | 15, 16, 17, 21 | subaddrii 10249 | . . . . . . 7 ⊢ (4 − 1) = 3 |
23 | df-3 10957 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
24 | 22, 23 | eqtri 2632 | . . . . . 6 ⊢ (4 − 1) = (2 + 1) |
25 | 24 | oveq2i 6560 | . . . . 5 ⊢ (2...(4 − 1)) = (2...(2 + 1)) |
26 | 2z 11286 | . . . . . 6 ⊢ 2 ∈ ℤ | |
27 | fzpr 12266 | . . . . . 6 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)}) | |
28 | 26, 27 | ax-mp 5 | . . . . 5 ⊢ (2...(2 + 1)) = {2, (2 + 1)} |
29 | 25, 28 | eqtri 2632 | . . . 4 ⊢ (2...(4 − 1)) = {2, (2 + 1)} |
30 | 23 | eqcomi 2619 | . . . . 5 ⊢ (2 + 1) = 3 |
31 | 30 | preq2i 4216 | . . . 4 ⊢ {2, (2 + 1)} = {2, 3} |
32 | 14, 29, 31 | 3eqtri 2636 | . . 3 ⊢ (2..^4) = {2, 3} |
33 | 11, 32 | uneq12i 3727 | . 2 ⊢ ((0..^2) ∪ (2..^4)) = ({0, 1} ∪ {2, 3}) |
34 | 10, 33 | eqtri 2632 | 1 ⊢ (0..^4) = ({0, 1} ∪ {2, 3}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {cpr 4127 class class class wbr 4583 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 2c2 10947 3c3 10948 4c4 10949 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 |
This theorem is referenced by: 4cycl4v4e 26194 4cycl4dv4e 26196 3pthdlem1 41331 upgr4cycl4dv4e 41352 |
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