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Mirrors > Home > MPE Home > Th. List > efgsval2 | Structured version Visualization version GIF version |
Description: Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 1-Oct-2015.) |
Ref | Expression |
---|---|
efgval.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
efgval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
efgval2.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
efgval2.t | ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) |
efgred.d | ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) |
efgred.s | ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) |
Ref | Expression |
---|---|
efgsval2 | ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
2 | efgval.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
3 | efgval2.m | . . . 4 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
4 | efgval2.t | . . . 4 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
5 | efgred.d | . . . 4 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | |
6 | efgred.s | . . . 4 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) | |
7 | 1, 2, 3, 4, 5, 6 | efgsval 19563 | . . 3 ⊢ ((𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆 → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = ((𝐴 ++ 〈“𝐵”〉)‘((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1))) |
8 | s1cl 14534 | . . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → 〈“𝐵”〉 ∈ Word 𝑊) | |
9 | ccatlen 14507 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 〈“𝐵”〉 ∈ Word 𝑊) → (♯‘(𝐴 ++ 〈“𝐵”〉)) = ((♯‘𝐴) + (♯‘〈“𝐵”〉))) | |
10 | 8, 9 | sylan2 593 | . . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (♯‘(𝐴 ++ 〈“𝐵”〉)) = ((♯‘𝐴) + (♯‘〈“𝐵”〉))) |
11 | s1len 14538 | . . . . . . . . 9 ⊢ (♯‘〈“𝐵”〉) = 1 | |
12 | 11 | oveq2i 7404 | . . . . . . . 8 ⊢ ((♯‘𝐴) + (♯‘〈“𝐵”〉)) = ((♯‘𝐴) + 1) |
13 | 10, 12 | eqtrdi 2787 | . . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (♯‘(𝐴 ++ 〈“𝐵”〉)) = ((♯‘𝐴) + 1)) |
14 | 13 | oveq1d 7408 | . . . . . 6 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1) = (((♯‘𝐴) + 1) − 1)) |
15 | lencl 14465 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ0) | |
16 | 15 | nn0cnd 12516 | . . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℂ) |
17 | ax-1cn 11150 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
18 | pncan 11448 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘𝐴) + 1) − 1) = (♯‘𝐴)) | |
19 | 16, 17, 18 | sylancl 586 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (((♯‘𝐴) + 1) − 1) = (♯‘𝐴)) |
20 | 16 | addlidd 11397 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (0 + (♯‘𝐴)) = (♯‘𝐴)) |
21 | 19, 20 | eqtr4d 2774 | . . . . . . 7 ⊢ (𝐴 ∈ Word 𝑊 → (((♯‘𝐴) + 1) − 1) = (0 + (♯‘𝐴))) |
22 | 21 | adantr 481 | . . . . . 6 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (((♯‘𝐴) + 1) − 1) = (0 + (♯‘𝐴))) |
23 | 14, 22 | eqtrd 2771 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1) = (0 + (♯‘𝐴))) |
24 | 23 | fveq2d 6882 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ++ 〈“𝐵”〉)‘((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1)) = ((𝐴 ++ 〈“𝐵”〉)‘(0 + (♯‘𝐴)))) |
25 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ Word 𝑊) | |
26 | 8 | adantl 482 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → 〈“𝐵”〉 ∈ Word 𝑊) |
27 | 1nn 12205 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
28 | 11, 27 | eqeltri 2828 | . . . . . . 7 ⊢ (♯‘〈“𝐵”〉) ∈ ℕ |
29 | lbfzo0 13654 | . . . . . . 7 ⊢ (0 ∈ (0..^(♯‘〈“𝐵”〉)) ↔ (♯‘〈“𝐵”〉) ∈ ℕ) | |
30 | 28, 29 | mpbir 230 | . . . . . 6 ⊢ 0 ∈ (0..^(♯‘〈“𝐵”〉)) |
31 | 30 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → 0 ∈ (0..^(♯‘〈“𝐵”〉))) |
32 | ccatval3 14511 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 〈“𝐵”〉 ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘〈“𝐵”〉))) → ((𝐴 ++ 〈“𝐵”〉)‘(0 + (♯‘𝐴))) = (〈“𝐵”〉‘0)) | |
33 | 25, 26, 31, 32 | syl3anc 1371 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ++ 〈“𝐵”〉)‘(0 + (♯‘𝐴))) = (〈“𝐵”〉‘0)) |
34 | s1fv 14542 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (〈“𝐵”〉‘0) = 𝐵) | |
35 | 34 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (〈“𝐵”〉‘0) = 𝐵) |
36 | 24, 33, 35 | 3eqtrd 2775 | . . 3 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ++ 〈“𝐵”〉)‘((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1)) = 𝐵) |
37 | 7, 36 | sylan9eqr 2793 | . 2 ⊢ (((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = 𝐵) |
38 | 37 | 3impa 1110 | 1 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 {crab 3431 ∖ cdif 3941 ∅c0 4318 {csn 4622 〈cop 4628 〈cotp 4630 ∪ ciun 4990 ↦ cmpt 5224 I cid 5566 × cxp 5667 dom cdm 5669 ran crn 5670 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 1oc1o 8441 2oc2o 8442 ℂcc 11090 0cc0 11092 1c1 11093 + caddc 11095 − cmin 11426 ℕcn 12194 ...cfz 13466 ..^cfzo 13609 ♯chash 14272 Word cword 14446 ++ cconcat 14502 〈“cs1 14527 splice csplice 14681 〈“cs2 14774 ~FG cefg 19538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-fzo 13610 df-hash 14273 df-word 14447 df-concat 14503 df-s1 14528 |
This theorem is referenced by: efgsfo 19571 efgredlemd 19576 efgrelexlemb 19582 |
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