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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > s3rnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of s2rn 15014 as of 1-Aug-2025. Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
s3rnOLD.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
s3rnOLD.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
s3rnOLD.k | ⊢ (𝜑 → 𝐾 ∈ 𝐷) |
Ref | Expression |
---|---|
s3rnOLD | ⊢ (𝜑 → ran 〈“𝐼𝐽𝐾”〉 = {𝐼, 𝐽, 𝐾}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmrn 6101 | . 2 ⊢ (〈“𝐼𝐽𝐾”〉 “ dom 〈“𝐼𝐽𝐾”〉) = ran 〈“𝐼𝐽𝐾”〉 | |
2 | s3rnOLD.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
3 | s3rnOLD.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
4 | s3rnOLD.k | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
5 | 2, 3, 4 | s3cld 14923 | . . . . . 6 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ Word 𝐷) |
6 | wrdfn 14578 | . . . . . 6 ⊢ (〈“𝐼𝐽𝐾”〉 ∈ Word 𝐷 → 〈“𝐼𝐽𝐾”〉 Fn (0..^(♯‘〈“𝐼𝐽𝐾”〉))) | |
7 | s3len 14945 | . . . . . . . . . 10 ⊢ (♯‘〈“𝐼𝐽𝐾”〉) = 3 | |
8 | 7 | oveq2i 7461 | . . . . . . . . 9 ⊢ (0..^(♯‘〈“𝐼𝐽𝐾”〉)) = (0..^3) |
9 | fzo0to3tp 13804 | . . . . . . . . 9 ⊢ (0..^3) = {0, 1, 2} | |
10 | 8, 9 | eqtri 2768 | . . . . . . . 8 ⊢ (0..^(♯‘〈“𝐼𝐽𝐾”〉)) = {0, 1, 2} |
11 | 10 | fneq2i 6679 | . . . . . . 7 ⊢ (〈“𝐼𝐽𝐾”〉 Fn (0..^(♯‘〈“𝐼𝐽𝐾”〉)) ↔ 〈“𝐼𝐽𝐾”〉 Fn {0, 1, 2}) |
12 | 11 | biimpi 216 | . . . . . 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 6686 | . . . 4 ⊢ (𝜑 → dom 〈“𝐼𝐽𝐾”〉 = {0, 1, 2}) |
15 | 14 | imaeq2d 6091 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉 “ dom 〈“𝐼𝐽𝐾”〉) = (〈“𝐼𝐽𝐾”〉 “ {0, 1, 2})) |
16 | c0ex 11286 | . . . . . 6 ⊢ 0 ∈ V | |
17 | 16 | tpid1 4793 | . . . . 5 ⊢ 0 ∈ {0, 1, 2} |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ {0, 1, 2}) |
19 | 1ex 11288 | . . . . . 6 ⊢ 1 ∈ V | |
20 | 19 | tpid2 4795 | . . . . 5 ⊢ 1 ∈ {0, 1, 2} |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ {0, 1, 2}) |
22 | 2ex 12372 | . . . . . 6 ⊢ 2 ∈ V | |
23 | 22 | tpid3 4798 | . . . . 5 ⊢ 2 ∈ {0, 1, 2} |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ {0, 1, 2}) |
25 | 13, 18, 21, 24 | fnimatpd 7008 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉 “ {0, 1, 2}) = {(〈“𝐼𝐽𝐾”〉‘0), (〈“𝐼𝐽𝐾”〉‘1), (〈“𝐼𝐽𝐾”〉‘2)}) |
26 | s3fv0 14942 | . . . . 5 ⊢ (𝐼 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘0) = 𝐼) | |
27 | 2, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘0) = 𝐼) |
28 | s3fv1 14943 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘1) = 𝐽) | |
29 | 3, 28 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘1) = 𝐽) |
30 | s3fv2 14944 | . . . . 5 ⊢ (𝐾 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘2) = 𝐾) | |
31 | 4, 30 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘2) = 𝐾) |
32 | 27, 29, 31 | tpeq123d 4773 | . . 3 ⊢ (𝜑 → {(〈“𝐼𝐽𝐾”〉‘0), (〈“𝐼𝐽𝐾”〉‘1), (〈“𝐼𝐽𝐾”〉‘2)} = {𝐼, 𝐽, 𝐾}) |
33 | 15, 25, 32 | 3eqtrd 2784 | . 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 1537 ∈ wcel 2108 {ctp 4652 dom cdm 5700 ran crn 5701 “ cima 5703 Fn wfn 6570 ‘cfv 6575 (class class class)co 7450 0cc0 11186 1c1 11187 2c2 12350 3c3 12351 ..^cfzo 13713 ♯chash 14381 Word cword 14564 〈“cs3 14893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-fzo 13714 df-hash 14382 df-word 14565 df-concat 14621 df-s1 14646 df-s2 14899 df-s3 14900 |
This theorem is referenced by: (None) |
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