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Mirrors > Home > MPE Home > Th. List > fzo13pr | Structured version Visualization version GIF version |
Description: A 1-based half-open integer interval up to, but not including, 3 is a pair. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
Ref | Expression |
---|---|
fzo13pr | ⊢ (1..^3) = {1, 2} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3z 12595 | . . . 4 ⊢ 3 ∈ ℤ | |
2 | fzoval 13633 | . . . 4 ⊢ (3 ∈ ℤ → (1..^3) = (1...(3 − 1))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (1..^3) = (1...(3 − 1)) |
4 | 3m1e2 12340 | . . . . 5 ⊢ (3 − 1) = 2 | |
5 | 1p1e2 12337 | . . . . 5 ⊢ (1 + 1) = 2 | |
6 | 4, 5 | eqtr4i 2764 | . . . 4 ⊢ (3 − 1) = (1 + 1) |
7 | 6 | oveq2i 7420 | . . 3 ⊢ (1...(3 − 1)) = (1...(1 + 1)) |
8 | 1z 12592 | . . . 4 ⊢ 1 ∈ ℤ | |
9 | fzpr 13556 | . . . 4 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
11 | 3, 7, 10 | 3eqtri 2765 | . 2 ⊢ (1..^3) = {1, (1 + 1)} |
12 | 5 | preq2i 4742 | . 2 ⊢ {1, (1 + 1)} = {1, 2} |
13 | 11, 12 | eqtri 2761 | 1 ⊢ (1..^3) = {1, 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cpr 4631 (class class class)co 7409 1c1 11111 + caddc 11113 − cmin 11444 2c2 12267 3c3 12268 ℤcz 12558 ...cfz 13484 ..^cfzo 13627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 |
This theorem is referenced by: istrkg3ld 27712 3pthdlem1 29417 lmat22det 32802 circlemethhgt 33655 |
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