Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 17967 | . . 3 ⊢ ((𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆 → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = ((𝐴 ++ 〈“𝐵”〉)‘((#‘(𝐴 ++ 〈“𝐵”〉)) − 1))) |
8 | 7 | 3ad2ant3 1077 | . 2 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = ((𝐴 ++ 〈“𝐵”〉)‘((#‘(𝐴 ++ 〈“𝐵”〉)) − 1))) |
9 | lencl 13179 | . . . . . . 7 ⊢ (𝐴 ∈ Word 𝑊 → (#‘𝐴) ∈ ℕ0) | |
10 | 9 | 3ad2ant1 1075 | . . . . . 6 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (#‘𝐴) ∈ ℕ0) |
11 | 10 | nn0cnd 11230 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (#‘𝐴) ∈ ℂ) |
12 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
13 | pncan 10166 | . . . . 5 ⊢ (((#‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((#‘𝐴) + 1) − 1) = (#‘𝐴)) | |
14 | 11, 12, 13 | sylancl 693 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (((#‘𝐴) + 1) − 1) = (#‘𝐴)) |
15 | simp1 1054 | . . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → 𝐴 ∈ Word 𝑊) | |
16 | simp2 1055 | . . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → 𝐵 ∈ 𝑊) | |
17 | 16 | s1cld 13236 | . . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → 〈“𝐵”〉 ∈ Word 𝑊) |
18 | ccatlen 13213 | . . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 〈“𝐵”〉 ∈ Word 𝑊) → (#‘(𝐴 ++ 〈“𝐵”〉)) = ((#‘𝐴) + (#‘〈“𝐵”〉))) | |
19 | 15, 17, 18 | syl2anc 691 | . . . . . 6 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (#‘(𝐴 ++ 〈“𝐵”〉)) = ((#‘𝐴) + (#‘〈“𝐵”〉))) |
20 | s1len 13238 | . . . . . . 7 ⊢ (#‘〈“𝐵”〉) = 1 | |
21 | 20 | oveq2i 6560 | . . . . . 6 ⊢ ((#‘𝐴) + (#‘〈“𝐵”〉)) = ((#‘𝐴) + 1) |
22 | 19, 21 | syl6eq 2660 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (#‘(𝐴 ++ 〈“𝐵”〉)) = ((#‘𝐴) + 1)) |
23 | 22 | oveq1d 6564 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → ((#‘(𝐴 ++ 〈“𝐵”〉)) − 1) = (((#‘𝐴) + 1) − 1)) |
24 | 11 | addid2d 10116 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (0 + (#‘𝐴)) = (#‘𝐴)) |
25 | 14, 23, 24 | 3eqtr4d 2654 | . . 3 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → ((#‘(𝐴 ++ 〈“𝐵”〉)) − 1) = (0 + (#‘𝐴))) |
26 | 25 | fveq2d 6107 | . 2 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → ((𝐴 ++ 〈“𝐵”〉)‘((#‘(𝐴 ++ 〈“𝐵”〉)) − 1)) = ((𝐴 ++ 〈“𝐵”〉)‘(0 + (#‘𝐴)))) |
27 | 1nn 10908 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
28 | 20, 27 | eqeltri 2684 | . . . . . 6 ⊢ (#‘〈“𝐵”〉) ∈ ℕ |
29 | 28 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (#‘〈“𝐵”〉) ∈ ℕ) |
30 | lbfzo0 12375 | . . . . 5 ⊢ (0 ∈ (0..^(#‘〈“𝐵”〉)) ↔ (#‘〈“𝐵”〉) ∈ ℕ) | |
31 | 29, 30 | sylibr 223 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → 0 ∈ (0..^(#‘〈“𝐵”〉))) |
32 | ccatval3 13216 | . . . 4 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 〈“𝐵”〉 ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘〈“𝐵”〉))) → ((𝐴 ++ 〈“𝐵”〉)‘(0 + (#‘𝐴))) = (〈“𝐵”〉‘0)) | |
33 | 15, 17, 31, 32 | syl3anc 1318 | . . 3 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → ((𝐴 ++ 〈“𝐵”〉)‘(0 + (#‘𝐴))) = (〈“𝐵”〉‘0)) |
34 | s1fv 13243 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (〈“𝐵”〉‘0) = 𝐵) | |
35 | 34 | 3ad2ant2 1076 | . . 3 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (〈“𝐵”〉‘0) = 𝐵) |
36 | 33, 35 | eqtrd 2644 | . 2 ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → ((𝐴 ++ 〈“𝐵”〉)‘(0 + (#‘𝐴))) = 𝐵) |
37 | 8, 26, 36 | 3eqtrd 2648 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∅c0 3874 {csn 4125 〈cop 4131 〈cotp 4133 ∪ ciun 4455 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1𝑜c1o 7440 2𝑜c2o 7441 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 〈“cs1 13149 splice csplice 13151 〈“cs2 13437 ~FG cefg 17942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 |
This theorem is referenced by: efgsfo 17975 efgredlemd 17980 efgrelexlemb 17986 |
Copyright terms: Public domain | W3C validator |