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Mirrors > Home > MPE Home > Th. List > s7rn | Structured version Visualization version GIF version |
Description: Range of a length 7 string. (Contributed by AV, 30-Jul-2025.) |
Ref | Expression |
---|---|
s7rn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
s7rn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
s7rn.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
s7rn.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
s7rn.e | ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
s7rn.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
s7rn.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
Ref | Expression |
---|---|
s7rn | ⊢ (𝜑 → ran 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s4s3 14976 | . . . 4 ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹𝐺”〉) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹𝐺”〉)) |
3 | 2 | rneqd 5962 | . 2 ⊢ (𝜑 → ran 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = ran (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹𝐺”〉)) |
4 | s4cli 14927 | . . . 4 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word V | |
5 | s3cli 14926 | . . . 4 ⊢ 〈“𝐸𝐹𝐺”〉 ∈ Word V | |
6 | 4, 5 | pm3.2i 470 | . . 3 ⊢ (〈“𝐴𝐵𝐶𝐷”〉 ∈ Word V ∧ 〈“𝐸𝐹𝐺”〉 ∈ Word V) |
7 | ccatrn 14633 | . . 3 ⊢ ((〈“𝐴𝐵𝐶𝐷”〉 ∈ Word V ∧ 〈“𝐸𝐹𝐺”〉 ∈ Word V) → ran (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹𝐺”〉) = (ran 〈“𝐴𝐵𝐶𝐷”〉 ∪ ran 〈“𝐸𝐹𝐺”〉)) | |
8 | 6, 7 | mp1i 13 | . 2 ⊢ (𝜑 → ran (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸𝐹𝐺”〉) = (ran 〈“𝐴𝐵𝐶𝐷”〉 ∪ ran 〈“𝐸𝐹𝐺”〉)) |
9 | df-s4 14895 | . . . . . 6 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉)) |
11 | 10 | rneqd 5962 | . . . 4 ⊢ (𝜑 → ran 〈“𝐴𝐵𝐶𝐷”〉 = ran (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉)) |
12 | s3cli 14926 | . . . . . 6 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
13 | s1cli 14649 | . . . . . 6 ⊢ 〈“𝐷”〉 ∈ Word V | |
14 | 12, 13 | pm3.2i 470 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word V ∧ 〈“𝐷”〉 ∈ Word V) |
15 | ccatrn 14633 | . . . . 5 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word V ∧ 〈“𝐷”〉 ∈ Word V) → ran (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) = (ran 〈“𝐴𝐵𝐶”〉 ∪ ran 〈“𝐷”〉)) | |
16 | 14, 15 | mp1i 13 | . . . 4 ⊢ (𝜑 → ran (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) = (ran 〈“𝐴𝐵𝐶”〉 ∪ ran 〈“𝐷”〉)) |
17 | s7rn.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
18 | s7rn.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
19 | s7rn.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
20 | 17, 18, 19 | s3rn 15009 | . . . . 5 ⊢ (𝜑 → ran 〈“𝐴𝐵𝐶”〉 = {𝐴, 𝐵, 𝐶}) |
21 | s7rn.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
22 | s1rn 14643 | . . . . . 6 ⊢ (𝐷 ∈ 𝑉 → ran 〈“𝐷”〉 = {𝐷}) | |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → ran 〈“𝐷”〉 = {𝐷}) |
24 | 20, 23 | uneq12d 4186 | . . . 4 ⊢ (𝜑 → (ran 〈“𝐴𝐵𝐶”〉 ∪ ran 〈“𝐷”〉) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})) |
25 | 11, 16, 24 | 3eqtrd 2778 | . . 3 ⊢ (𝜑 → ran 〈“𝐴𝐵𝐶𝐷”〉 = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})) |
26 | s7rn.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉) | |
27 | s7rn.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
28 | s7rn.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
29 | 26, 27, 28 | s3rn 15009 | . . 3 ⊢ (𝜑 → ran 〈“𝐸𝐹𝐺”〉 = {𝐸, 𝐹, 𝐺}) |
30 | 25, 29 | uneq12d 4186 | . 2 ⊢ (𝜑 → (ran 〈“𝐴𝐵𝐶𝐷”〉 ∪ ran 〈“𝐸𝐹𝐺”〉) = (({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})) |
31 | 3, 8, 30 | 3eqtrd 2778 | 1 ⊢ (𝜑 → ran 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ∪ cun 3968 {csn 4648 {ctp 4652 ran crn 5700 (class class class)co 7445 Word cword 14558 ++ cconcat 14614 〈“cs1 14639 〈“cs3 14887 〈“cs4 14888 〈“cs7 14891 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-hash 14376 df-word 14559 df-concat 14615 df-s1 14640 df-s2 14893 df-s3 14894 df-s4 14895 df-s5 14896 df-s6 14897 df-s7 14898 |
This theorem is referenced by: s7f1o 15011 usgrexmpl1edg 47759 usgrexmpl2edg 47764 |
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