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Mirrors > Home > MPE Home > Th. List > eqs1 | Structured version Visualization version GIF version |
Description: A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.) |
Ref | Expression |
---|---|
eqs1 | ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 𝑊 = ⟨“(𝑊‘0)”⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ ((♯‘𝑊) = 1 → (♯‘𝑊) = 1) | |
2 | s1len 14582 | . . . . 5 ⊢ (♯‘⟨“(𝑊‘0)”⟩) = 1 | |
3 | 1, 2 | eqtr4di 2786 | . . . 4 ⊢ ((♯‘𝑊) = 1 → (♯‘𝑊) = (♯‘⟨“(𝑊‘0)”⟩)) |
4 | fvex 6904 | . . . . . . . 8 ⊢ (𝑊‘0) ∈ V | |
5 | s1fv 14586 | . . . . . . . 8 ⊢ ((𝑊‘0) ∈ V → (⟨“(𝑊‘0)”⟩‘0) = (𝑊‘0)) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (⟨“(𝑊‘0)”⟩‘0) = (𝑊‘0) |
7 | 6 | eqcomi 2737 | . . . . . 6 ⊢ (𝑊‘0) = (⟨“(𝑊‘0)”⟩‘0) |
8 | c0ex 11232 | . . . . . . 7 ⊢ 0 ∈ V | |
9 | fveq2 6891 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑊‘𝑥) = (𝑊‘0)) | |
10 | fveq2 6891 | . . . . . . . 8 ⊢ (𝑥 = 0 → (⟨“(𝑊‘0)”⟩‘𝑥) = (⟨“(𝑊‘0)”⟩‘0)) | |
11 | 9, 10 | eqeq12d 2744 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) ↔ (𝑊‘0) = (⟨“(𝑊‘0)”⟩‘0))) |
12 | 8, 11 | ralsn 4681 | . . . . . 6 ⊢ (∀𝑥 ∈ {0} (𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) ↔ (𝑊‘0) = (⟨“(𝑊‘0)”⟩‘0)) |
13 | 7, 12 | mpbir 230 | . . . . 5 ⊢ ∀𝑥 ∈ {0} (𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) |
14 | oveq2 7422 | . . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1)) | |
15 | fzo01 13740 | . . . . . . 7 ⊢ (0..^1) = {0} | |
16 | 14, 15 | eqtrdi 2784 | . . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0}) |
17 | 16 | raleqdv 3321 | . . . . 5 ⊢ ((♯‘𝑊) = 1 → (∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥) ↔ ∀𝑥 ∈ {0} (𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥))) |
18 | 13, 17 | mpbiri 258 | . . . 4 ⊢ ((♯‘𝑊) = 1 → ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥)) |
19 | 3, 18 | jca 511 | . . 3 ⊢ ((♯‘𝑊) = 1 → ((♯‘𝑊) = (♯‘⟨“(𝑊‘0)”⟩) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥))) |
20 | s1cli 14581 | . . . 4 ⊢ ⟨“(𝑊‘0)”⟩ ∈ Word V | |
21 | eqwrd 14533 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ ⟨“(𝑊‘0)”⟩ ∈ Word V) → (𝑊 = ⟨“(𝑊‘0)”⟩ ↔ ((♯‘𝑊) = (♯‘⟨“(𝑊‘0)”⟩) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥)))) | |
22 | 20, 21 | mpan2 690 | . . 3 ⊢ (𝑊 ∈ Word 𝐴 → (𝑊 = ⟨“(𝑊‘0)”⟩ ↔ ((♯‘𝑊) = (♯‘⟨“(𝑊‘0)”⟩) ∧ ∀𝑥 ∈ (0..^(♯‘𝑊))(𝑊‘𝑥) = (⟨“(𝑊‘0)”⟩‘𝑥)))) |
23 | 19, 22 | imbitrrid 245 | . 2 ⊢ (𝑊 ∈ Word 𝐴 → ((♯‘𝑊) = 1 → 𝑊 = ⟨“(𝑊‘0)”⟩)) |
24 | 23 | imp 406 | 1 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (♯‘𝑊) = 1) → 𝑊 = ⟨“(𝑊‘0)”⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 {csn 4624 ‘cfv 6542 (class class class)co 7414 0cc0 11132 1c1 11133 ..^cfzo 13653 ♯chash 14315 Word cword 14490 ⟨“cs1 14571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 df-hash 14316 df-word 14491 df-s1 14572 |
This theorem is referenced by: wrdl1exs1 14589 wrdl1s1 14590 swrds1 14642 revs1 14741 signsvtn0 34196 |
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