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Mirrors > Home > MPE Home > Th. List > s2prop | Structured version Visualization version GIF version |
Description: A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.) |
Ref | Expression |
---|---|
s2prop | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s2 13639 | . 2 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | s1cl 13418 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 ∈ Word 𝑆) | |
3 | cats1un 13521 | . . . 4 ⊢ ((〈“𝐴”〉 ∈ Word 𝑆 ∧ 𝐵 ∈ 𝑆) → (〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝐴”〉 ∪ {〈(#‘〈“𝐴”〉), 𝐵〉})) | |
4 | 2, 3 | sylan 487 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝐴”〉 ∪ {〈(#‘〈“𝐴”〉), 𝐵〉})) |
5 | s1val 13414 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴”〉 = {〈0, 𝐴〉}) |
7 | 6 | uneq1d 3799 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (〈“𝐴”〉 ∪ {〈(#‘〈“𝐴”〉), 𝐵〉}) = ({〈0, 𝐴〉} ∪ {〈(#‘〈“𝐴”〉), 𝐵〉})) |
8 | df-pr 4213 | . . . 4 ⊢ {〈0, 𝐴〉, 〈(#‘〈“𝐴”〉), 𝐵〉} = ({〈0, 𝐴〉} ∪ {〈(#‘〈“𝐴”〉), 𝐵〉}) | |
9 | s1len 13422 | . . . . . . 7 ⊢ (#‘〈“𝐴”〉) = 1 | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (#‘〈“𝐴”〉) = 1) |
11 | 10 | opeq1d 4439 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈(#‘〈“𝐴”〉), 𝐵〉 = 〈1, 𝐵〉) |
12 | 11 | preq2d 4307 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {〈0, 𝐴〉, 〈(#‘〈“𝐴”〉), 𝐵〉} = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
13 | 8, 12 | syl5eqr 2699 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ({〈0, 𝐴〉} ∪ {〈(#‘〈“𝐴”〉), 𝐵〉}) = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
14 | 4, 7, 13 | 3eqtrd 2689 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (〈“𝐴”〉 ++ 〈“𝐵”〉) = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
15 | 1, 14 | syl5eq 2697 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∪ cun 3605 {csn 4210 {cpr 4212 〈cop 4216 ‘cfv 5926 (class class class)co 6690 0cc0 9974 1c1 9975 #chash 13157 Word cword 13323 ++ cconcat 13325 〈“cs1 13326 〈“cs2 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-concat 13333 df-s1 13334 df-s2 13639 |
This theorem is referenced by: s2dmALT 13699 s3tpop 13700 s4prop 13701 funcnvs2 13704 s2f1o 13707 wrdlen2s2 13735 uhgrwkspthlem2 26706 ntrl2v2e 27136 |
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