Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > s3rn | Structured version Visualization version GIF version |
Description: Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
Ref | Expression |
---|---|
s3rn.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
s3rn.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
s3rn.k | ⊢ (𝜑 → 𝐾 ∈ 𝐷) |
Ref | Expression |
---|---|
s3rn | ⊢ (𝜑 → ran 〈“𝐼𝐽𝐾”〉 = {𝐼, 𝐽, 𝐾}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmrn 5956 | . 2 ⊢ (〈“𝐼𝐽𝐾”〉 “ dom 〈“𝐼𝐽𝐾”〉) = ran 〈“𝐼𝐽𝐾”〉 | |
2 | s3rn.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
3 | s3rn.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
4 | s3rn.k | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
5 | 2, 3, 4 | s3cld 14469 | . . . . . 6 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ Word 𝐷) |
6 | wrdfn 14115 | . . . . . 6 ⊢ (〈“𝐼𝐽𝐾”〉 ∈ Word 𝐷 → 〈“𝐼𝐽𝐾”〉 Fn (0..^(♯‘〈“𝐼𝐽𝐾”〉))) | |
7 | s3len 14491 | . . . . . . . . . 10 ⊢ (♯‘〈“𝐼𝐽𝐾”〉) = 3 | |
8 | 7 | oveq2i 7245 | . . . . . . . . 9 ⊢ (0..^(♯‘〈“𝐼𝐽𝐾”〉)) = (0..^3) |
9 | fzo0to3tp 13357 | . . . . . . . . 9 ⊢ (0..^3) = {0, 1, 2} | |
10 | 8, 9 | eqtri 2767 | . . . . . . . 8 ⊢ (0..^(♯‘〈“𝐼𝐽𝐾”〉)) = {0, 1, 2} |
11 | 10 | fneq2i 6497 | . . . . . . 7 ⊢ (〈“𝐼𝐽𝐾”〉 Fn (0..^(♯‘〈“𝐼𝐽𝐾”〉)) ↔ 〈“𝐼𝐽𝐾”〉 Fn {0, 1, 2}) |
12 | 11 | biimpi 219 | . . . . . 6 ⊢ (〈“𝐼𝐽𝐾”〉 Fn (0..^(♯‘〈“𝐼𝐽𝐾”〉)) → 〈“𝐼𝐽𝐾”〉 Fn {0, 1, 2}) |
13 | 5, 6, 12 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 Fn {0, 1, 2}) |
14 | 13 | fndmd 6504 | . . . 4 ⊢ (𝜑 → dom 〈“𝐼𝐽𝐾”〉 = {0, 1, 2}) |
15 | 14 | imaeq2d 5946 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉 “ dom 〈“𝐼𝐽𝐾”〉) = (〈“𝐼𝐽𝐾”〉 “ {0, 1, 2})) |
16 | c0ex 10856 | . . . . . 6 ⊢ 0 ∈ V | |
17 | 16 | tpid1 4700 | . . . . 5 ⊢ 0 ∈ {0, 1, 2} |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ {0, 1, 2}) |
19 | 1ex 10858 | . . . . . 6 ⊢ 1 ∈ V | |
20 | 19 | tpid2 4702 | . . . . 5 ⊢ 1 ∈ {0, 1, 2} |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ {0, 1, 2}) |
22 | 2ex 11936 | . . . . . 6 ⊢ 2 ∈ V | |
23 | 22 | tpid3 4705 | . . . . 5 ⊢ 2 ∈ {0, 1, 2} |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ {0, 1, 2}) |
25 | 13, 18, 21, 24 | fnimatp 30765 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉 “ {0, 1, 2}) = {(〈“𝐼𝐽𝐾”〉‘0), (〈“𝐼𝐽𝐾”〉‘1), (〈“𝐼𝐽𝐾”〉‘2)}) |
26 | s3fv0 14488 | . . . . 5 ⊢ (𝐼 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘0) = 𝐼) | |
27 | 2, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘0) = 𝐼) |
28 | s3fv1 14489 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘1) = 𝐽) | |
29 | 3, 28 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘1) = 𝐽) |
30 | s3fv2 14490 | . . . . 5 ⊢ (𝐾 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘2) = 𝐾) | |
31 | 4, 30 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘2) = 𝐾) |
32 | 27, 29, 31 | tpeq123d 4680 | . . 3 ⊢ (𝜑 → {(〈“𝐼𝐽𝐾”〉‘0), (〈“𝐼𝐽𝐾”〉‘1), (〈“𝐼𝐽𝐾”〉‘2)} = {𝐼, 𝐽, 𝐾}) |
33 | 15, 25, 32 | 3eqtrd 2783 | . 2 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉 “ dom 〈“𝐼𝐽𝐾”〉) = {𝐼, 𝐽, 𝐾}) |
34 | 1, 33 | eqtr3id 2794 | 1 ⊢ (𝜑 → ran 〈“𝐼𝐽𝐾”〉 = {𝐼, 𝐽, 𝐾}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 {ctp 4561 dom cdm 5568 ran crn 5569 “ cima 5571 Fn wfn 6395 ‘cfv 6400 (class class class)co 7234 0cc0 10758 1c1 10759 2c2 11914 3c3 11915 ..^cfzo 13267 ♯chash 13928 Word cword 14101 〈“cs3 14439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-int 4876 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-card 9584 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-nn 11860 df-2 11922 df-3 11923 df-n0 12120 df-z 12206 df-uz 12468 df-fz 13125 df-fzo 13268 df-hash 13929 df-word 14102 df-concat 14158 df-s1 14185 df-s2 14445 df-s3 14446 |
This theorem is referenced by: cyc3co2 31157 |
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