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Mirrors > Home > MPE Home > Th. List > s3tpop | Structured version Visualization version GIF version |
Description: A length 3 word is an unordered triple of ordered pairs. (Contributed by AV, 23-Jan-2021.) |
Ref | Expression |
---|---|
s3tpop | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 〈“𝐴𝐵𝐶”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s3 14543 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
2 | s2cl 14572 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 ∈ Word 𝑆) | |
3 | cats1un 14415 | . . . 4 ⊢ ((〈“𝐴𝐵”〉 ∈ Word 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝐴𝐵”〉 ∪ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉})) | |
4 | 2, 3 | stoic3 1782 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝐴𝐵”〉 ∪ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉})) |
5 | s2prop 14601 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) | |
6 | 5 | 3adant3 1130 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
7 | s2len 14583 | . . . . . . 7 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
8 | 7 | opeq1i 4812 | . . . . . 6 ⊢ 〈(♯‘〈“𝐴𝐵”〉), 𝐶〉 = 〈2, 𝐶〉 |
9 | 8 | sneqi 4577 | . . . . 5 ⊢ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉} = {〈2, 𝐶〉} |
10 | 9 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉} = {〈2, 𝐶〉}) |
11 | 6, 10 | uneq12d 4102 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ∪ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉}) = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉})) |
12 | df-tp 4571 | . . . . 5 ⊢ {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉} = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉}) | |
13 | 12 | eqcomi 2748 | . . . 4 ⊢ ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉}) = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉} |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉}) = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉}) |
15 | 4, 11, 14 | 3eqtrd 2783 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉}) |
16 | 1, 15 | eqtrid 2791 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 〈“𝐴𝐵𝐶”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∪ cun 3889 {csn 4566 {cpr 4568 {ctp 4570 〈cop 4572 ‘cfv 6430 (class class class)co 7268 0cc0 10855 1c1 10856 2c2 12011 ♯chash 14025 Word cword 14198 ++ cconcat 14254 〈“cs1 14281 〈“cs2 14535 〈“cs3 14536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-fzo 13365 df-hash 14026 df-word 14199 df-concat 14255 df-s1 14282 df-s2 14542 df-s3 14543 |
This theorem is referenced by: funcnvs3 14608 wrdlen3s3 14643 |
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