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Mirrors > Home > MPE Home > Th. List > fzo1to4tp | Structured version Visualization version GIF version |
Description: A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.) |
Ref | Expression |
---|---|
fzo1to4tp | ⊢ (1..^4) = {1, 2, 3} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4z 12600 | . . 3 ⊢ 4 ∈ ℤ | |
2 | fzoval 13639 | . . 3 ⊢ (4 ∈ ℤ → (1..^4) = (1...(4 − 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (1..^4) = (1...(4 − 1)) |
4 | 4m1e3 12345 | . . . 4 ⊢ (4 − 1) = 3 | |
5 | df-3 12280 | . . . 4 ⊢ 3 = (2 + 1) | |
6 | 2cn 12291 | . . . . 5 ⊢ 2 ∈ ℂ | |
7 | ax-1cn 11170 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | 6, 7 | addcomi 11409 | . . . 4 ⊢ (2 + 1) = (1 + 2) |
9 | 4, 5, 8 | 3eqtri 2758 | . . 3 ⊢ (4 − 1) = (1 + 2) |
10 | 9 | oveq2i 7416 | . 2 ⊢ (1...(4 − 1)) = (1...(1 + 2)) |
11 | 1z 12596 | . . 3 ⊢ 1 ∈ ℤ | |
12 | fztp 13563 | . . . 4 ⊢ (1 ∈ ℤ → (1...(1 + 2)) = {1, (1 + 1), (1 + 2)}) | |
13 | eqidd 2727 | . . . . 5 ⊢ (1 ∈ ℤ → 1 = 1) | |
14 | 1p1e2 12341 | . . . . . 6 ⊢ (1 + 1) = 2 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (1 ∈ ℤ → (1 + 1) = 2) |
16 | 1p2e3 12359 | . . . . . 6 ⊢ (1 + 2) = 3 | |
17 | 16 | a1i 11 | . . . . 5 ⊢ (1 ∈ ℤ → (1 + 2) = 3) |
18 | 13, 15, 17 | tpeq123d 4747 | . . . 4 ⊢ (1 ∈ ℤ → {1, (1 + 1), (1 + 2)} = {1, 2, 3}) |
19 | 12, 18 | eqtrd 2766 | . . 3 ⊢ (1 ∈ ℤ → (1...(1 + 2)) = {1, 2, 3}) |
20 | 11, 19 | ax-mp 5 | . 2 ⊢ (1...(1 + 2)) = {1, 2, 3} |
21 | 3, 10, 20 | 3eqtri 2758 | 1 ⊢ (1..^4) = {1, 2, 3} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {ctp 4627 (class class class)co 7405 1c1 11113 + caddc 11115 − cmin 11448 2c2 12271 3c3 12272 4c4 12273 ℤcz 12562 ...cfz 13490 ..^cfzo 13633 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 |
This theorem is referenced by: fmtno4prmfac 46812 fmtnofz04prm 46817 |
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