| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 14792 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
| 2 | s2cl 14821 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 ∈ Word 𝑆) | |
| 3 | cats1un 14663 | . . . 4 ⊢ ((〈“𝐴𝐵”〉 ∈ Word 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝐴𝐵”〉 ∪ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉})) | |
| 4 | 2, 3 | stoic3 1776 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝐴𝐵”〉 ∪ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉})) |
| 5 | s2prop 14850 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) | |
| 6 | 5 | 3adant3 1132 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
| 7 | s2len 14832 | . . . . . . 7 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
| 8 | 7 | opeq1i 4836 | . . . . . 6 ⊢ 〈(♯‘〈“𝐴𝐵”〉), 𝐶〉 = 〈2, 𝐶〉 |
| 9 | 8 | sneqi 4596 | . . . . 5 ⊢ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉} = {〈2, 𝐶〉} |
| 10 | 9 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉} = {〈2, 𝐶〉}) |
| 11 | 6, 10 | uneq12d 4128 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ∪ {〈(♯‘〈“𝐴𝐵”〉), 𝐶〉}) = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉})) |
| 12 | df-tp 4590 | . . . . 5 ⊢ {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉} = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉}) | |
| 13 | 12 | eqcomi 2738 | . . . 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 2768 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉}) |
| 16 | 1, 15 | eqtrid 2776 | 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 1086 = wceq 1540 ∈ wcel 2109 ∪ cun 3909 {csn 4585 {cpr 4587 {ctp 4589 〈cop 4591 ‘cfv 6499 (class class class)co 7369 0cc0 11046 1c1 11047 2c2 12219 ♯chash 14273 Word cword 14456 ++ cconcat 14513 〈“cs1 14538 〈“cs2 14784 〈“cs3 14785 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11102 ax-resscn 11103 ax-1cn 11104 ax-icn 11105 ax-addcl 11106 ax-addrcl 11107 ax-mulcl 11108 ax-mulrcl 11109 ax-mulcom 11110 ax-addass 11111 ax-mulass 11112 ax-distr 11113 ax-i2m1 11114 ax-1ne0 11115 ax-1rid 11116 ax-rnegex 11117 ax-rrecex 11118 ax-cnre 11119 ax-pre-lttri 11120 ax-pre-lttrn 11121 ax-pre-ltadd 11122 ax-pre-mulgt0 11123 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9870 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11385 df-neg 11386 df-nn 12165 df-2 12227 df-n0 12421 df-z 12508 df-uz 12772 df-fz 13447 df-fzo 13594 df-hash 14274 df-word 14457 df-concat 14514 df-s1 14539 df-s2 14791 df-s3 14792 |
| This theorem is referenced by: funcnvs3 14857 wrdlen3s3 14892 |
| Copyright terms: Public domain | W3C validator |