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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 18859 | . . 3 ⊢ ((𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆 → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = ((𝐴 ++ 〈“𝐵”〉)‘((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1))) |
8 | s1cl 13958 | . . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → 〈“𝐵”〉 ∈ Word 𝑊) | |
9 | ccatlen 13929 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 〈“𝐵”〉 ∈ Word 𝑊) → (♯‘(𝐴 ++ 〈“𝐵”〉)) = ((♯‘𝐴) + (♯‘〈“𝐵”〉))) | |
10 | 8, 9 | sylan2 594 | . . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (♯‘(𝐴 ++ 〈“𝐵”〉)) = ((♯‘𝐴) + (♯‘〈“𝐵”〉))) |
11 | s1len 13962 | . . . . . . . . 9 ⊢ (♯‘〈“𝐵”〉) = 1 | |
12 | 11 | oveq2i 7169 | . . . . . . . 8 ⊢ ((♯‘𝐴) + (♯‘〈“𝐵”〉)) = ((♯‘𝐴) + 1) |
13 | 10, 12 | syl6eq 2874 | . . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (♯‘(𝐴 ++ 〈“𝐵”〉)) = ((♯‘𝐴) + 1)) |
14 | 13 | oveq1d 7173 | . . . . . 6 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1) = (((♯‘𝐴) + 1) − 1)) |
15 | lencl 13885 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ0) | |
16 | 15 | nn0cnd 11960 | . . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℂ) |
17 | ax-1cn 10597 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
18 | pncan 10894 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘𝐴) + 1) − 1) = (♯‘𝐴)) | |
19 | 16, 17, 18 | sylancl 588 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (((♯‘𝐴) + 1) − 1) = (♯‘𝐴)) |
20 | 16 | addid2d 10843 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (0 + (♯‘𝐴)) = (♯‘𝐴)) |
21 | 19, 20 | eqtr4d 2861 | . . . . . . 7 ⊢ (𝐴 ∈ Word 𝑊 → (((♯‘𝐴) + 1) − 1) = (0 + (♯‘𝐴))) |
22 | 21 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (((♯‘𝐴) + 1) − 1) = (0 + (♯‘𝐴))) |
23 | 14, 22 | eqtrd 2858 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1) = (0 + (♯‘𝐴))) |
24 | 23 | fveq2d 6676 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ++ 〈“𝐵”〉)‘((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1)) = ((𝐴 ++ 〈“𝐵”〉)‘(0 + (♯‘𝐴)))) |
25 | simpl 485 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ Word 𝑊) | |
26 | 8 | adantl 484 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → 〈“𝐵”〉 ∈ Word 𝑊) |
27 | 1nn 11651 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
28 | 11, 27 | eqeltri 2911 | . . . . . . 7 ⊢ (♯‘〈“𝐵”〉) ∈ ℕ |
29 | lbfzo0 13080 | . . . . . . 7 ⊢ (0 ∈ (0..^(♯‘〈“𝐵”〉)) ↔ (♯‘〈“𝐵”〉) ∈ ℕ) | |
30 | 28, 29 | mpbir 233 | . . . . . 6 ⊢ 0 ∈ (0..^(♯‘〈“𝐵”〉)) |
31 | 30 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → 0 ∈ (0..^(♯‘〈“𝐵”〉))) |
32 | ccatval3 13935 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 〈“𝐵”〉 ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘〈“𝐵”〉))) → ((𝐴 ++ 〈“𝐵”〉)‘(0 + (♯‘𝐴))) = (〈“𝐵”〉‘0)) | |
33 | 25, 26, 31, 32 | syl3anc 1367 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ++ 〈“𝐵”〉)‘(0 + (♯‘𝐴))) = (〈“𝐵”〉‘0)) |
34 | s1fv 13966 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (〈“𝐵”〉‘0) = 𝐵) | |
35 | 34 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → (〈“𝐵”〉‘0) = 𝐵) |
36 | 24, 33, 35 | 3eqtrd 2862 | . . 3 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ++ 〈“𝐵”〉)‘((♯‘(𝐴 ++ 〈“𝐵”〉)) − 1)) = 𝐵) |
37 | 7, 36 | sylan9eqr 2880 | . 2 ⊢ (((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊) ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = 𝐵) |
38 | 37 | 3impa 1106 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ∖ cdif 3935 ∅c0 4293 {csn 4569 〈cop 4575 〈cotp 4577 ∪ ciun 4921 ↦ cmpt 5148 I cid 5461 × cxp 5555 dom cdm 5557 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1oc1o 8097 2oc2o 8098 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 − cmin 10872 ℕcn 11640 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 ++ cconcat 13924 〈“cs1 13951 splice csplice 14113 〈“cs2 14205 ~FG cefg 18834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 |
This theorem is referenced by: efgsfo 18867 efgredlemd 18872 efgrelexlemb 18878 |
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