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Mirrors > Home > ILE Home > Th. List > fzo0to42pr | 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 9196 | . . . 4 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 9198 | . . . 4 ⊢ 4 ∈ ℕ0 | |
3 | 2re 8992 | . . . . 5 ⊢ 2 ∈ ℝ | |
4 | 4re 8999 | . . . . 5 ⊢ 4 ∈ ℝ | |
5 | 2lt4 9095 | . . . . 5 ⊢ 2 < 4 | |
6 | 3, 4, 5 | ltleii 8063 | . . . 4 ⊢ 2 ≤ 4 |
7 | elfz2nn0 10115 | . . . 4 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ0 ∧ 4 ∈ ℕ0 ∧ 2 ≤ 4)) | |
8 | 1, 2, 6, 7 | mpbir3an 1179 | . . 3 ⊢ 2 ∈ (0...4) |
9 | fzosplit 10180 | . . 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 10221 | . . 3 ⊢ (0..^2) = {0, 1} | |
12 | 4z 9286 | . . . . 5 ⊢ 4 ∈ ℤ | |
13 | fzoval 10151 | . . . . 5 ⊢ (4 ∈ ℤ → (2..^4) = (2...(4 − 1))) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (2..^4) = (2...(4 − 1)) |
15 | 4cn 9000 | . . . . . . . 8 ⊢ 4 ∈ ℂ | |
16 | ax-1cn 7907 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
17 | 3cn 8997 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
18 | df-4 8983 | . . . . . . . . . 10 ⊢ 4 = (3 + 1) | |
19 | 17, 16 | addcomi 8104 | . . . . . . . . . 10 ⊢ (3 + 1) = (1 + 3) |
20 | 18, 19 | eqtri 2198 | . . . . . . . . 9 ⊢ 4 = (1 + 3) |
21 | 20 | eqcomi 2181 | . . . . . . . 8 ⊢ (1 + 3) = 4 |
22 | 15, 16, 17, 21 | subaddrii 8249 | . . . . . . 7 ⊢ (4 − 1) = 3 |
23 | df-3 8982 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
24 | 22, 23 | eqtri 2198 | . . . . . 6 ⊢ (4 − 1) = (2 + 1) |
25 | 24 | oveq2i 5889 | . . . . 5 ⊢ (2...(4 − 1)) = (2...(2 + 1)) |
26 | 2z 9284 | . . . . . 6 ⊢ 2 ∈ ℤ | |
27 | fzpr 10080 | . . . . . 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 2198 | . . . 4 ⊢ (2...(4 − 1)) = {2, (2 + 1)} |
30 | 23 | eqcomi 2181 | . . . . 5 ⊢ (2 + 1) = 3 |
31 | 30 | preq2i 3675 | . . . 4 ⊢ {2, (2 + 1)} = {2, 3} |
32 | 14, 29, 31 | 3eqtri 2202 | . . 3 ⊢ (2..^4) = {2, 3} |
33 | 11, 32 | uneq12i 3289 | . 2 ⊢ ((0..^2) ∪ (2..^4)) = ({0, 1} ∪ {2, 3}) |
34 | 10, 33 | eqtri 2198 | 1 ⊢ (0..^4) = ({0, 1} ∪ {2, 3}) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ∪ cun 3129 {cpr 3595 class class class wbr 4005 (class class class)co 5878 0cc0 7814 1c1 7815 + caddc 7817 ≤ cle 7996 − cmin 8131 2c2 8973 3c3 8974 4c4 8975 ℕ0cn0 9179 ℤcz 9256 ...cfz 10011 ..^cfzo 10145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-fz 10012 df-fzo 10146 |
This theorem is referenced by: (None) |
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