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Mirrors > Home > MPE Home > Th. List > wlk2v2elem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for wlk2v2e 27017: The values of 𝐼 after 𝐹 are edges between two vertices enumerated by 𝑃. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
Ref | Expression |
---|---|
wlk2v2e.i | ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 |
wlk2v2e.f | ⊢ 𝐹 = 〈“00”〉 |
wlk2v2e.x | ⊢ 𝑋 ∈ V |
wlk2v2e.y | ⊢ 𝑌 ∈ V |
wlk2v2e.p | ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 |
Ref | Expression |
---|---|
wlk2v2elem2 | ⊢ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
2 | 1 | fveq1i 6192 | . . . . . 6 ⊢ (𝐹‘0) = (〈“00”〉‘0) |
3 | 0z 11388 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
4 | s2fv0 13632 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘0) = 0) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘0) = 0 |
6 | 2, 5 | eqtri 2644 | . . . . 5 ⊢ (𝐹‘0) = 0 |
7 | 6 | fveq2i 6194 | . . . 4 ⊢ (𝐼‘(𝐹‘0)) = (𝐼‘0) |
8 | wlk2v2e.i | . . . . . 6 ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 | |
9 | 8 | fveq1i 6192 | . . . . 5 ⊢ (𝐼‘0) = (〈“{𝑋, 𝑌}”〉‘0) |
10 | prex 4909 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
11 | s1fv 13390 | . . . . . 6 ⊢ ({𝑋, 𝑌} ∈ V → (〈“{𝑋, 𝑌}”〉‘0) = {𝑋, 𝑌}) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ (〈“{𝑋, 𝑌}”〉‘0) = {𝑋, 𝑌} |
13 | 9, 12 | eqtri 2644 | . . . 4 ⊢ (𝐼‘0) = {𝑋, 𝑌} |
14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
15 | 14 | fveq1i 6192 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝑋𝑌𝑋”〉‘0) |
16 | wlk2v2e.x | . . . . . . . 8 ⊢ 𝑋 ∈ V | |
17 | s3fv0 13636 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘0) = 𝑋) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘0) = 𝑋 |
19 | 15, 18 | eqtri 2644 | . . . . . 6 ⊢ (𝑃‘0) = 𝑋 |
20 | 14 | fveq1i 6192 | . . . . . . 7 ⊢ (𝑃‘1) = (〈“𝑋𝑌𝑋”〉‘1) |
21 | wlk2v2e.y | . . . . . . . 8 ⊢ 𝑌 ∈ V | |
22 | s3fv1 13637 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (〈“𝑋𝑌𝑋”〉‘1) = 𝑌) | |
23 | 21, 22 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘1) = 𝑌 |
24 | 20, 23 | eqtri 2644 | . . . . . 6 ⊢ (𝑃‘1) = 𝑌 |
25 | 19, 24 | preq12i 4273 | . . . . 5 ⊢ {(𝑃‘0), (𝑃‘1)} = {𝑋, 𝑌} |
26 | 25 | eqcomi 2631 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘0), (𝑃‘1)} |
27 | 7, 13, 26 | 3eqtri 2648 | . . 3 ⊢ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} |
28 | 1 | fveq1i 6192 | . . . . . 6 ⊢ (𝐹‘1) = (〈“00”〉‘1) |
29 | s2fv1 13633 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘1) = 0) | |
30 | 3, 29 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘1) = 0 |
31 | 28, 30 | eqtri 2644 | . . . . 5 ⊢ (𝐹‘1) = 0 |
32 | 31 | fveq2i 6194 | . . . 4 ⊢ (𝐼‘(𝐹‘1)) = (𝐼‘0) |
33 | prcom 4267 | . . . . 5 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
34 | 14 | fveq1i 6192 | . . . . . . . 8 ⊢ (𝑃‘2) = (〈“𝑋𝑌𝑋”〉‘2) |
35 | s3fv2 13638 | . . . . . . . . 9 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘2) = 𝑋) | |
36 | 16, 35 | ax-mp 5 | . . . . . . . 8 ⊢ (〈“𝑋𝑌𝑋”〉‘2) = 𝑋 |
37 | 34, 36 | eqtri 2644 | . . . . . . 7 ⊢ (𝑃‘2) = 𝑋 |
38 | 24, 37 | preq12i 4273 | . . . . . 6 ⊢ {(𝑃‘1), (𝑃‘2)} = {𝑌, 𝑋} |
39 | 38 | eqcomi 2631 | . . . . 5 ⊢ {𝑌, 𝑋} = {(𝑃‘1), (𝑃‘2)} |
40 | 33, 39 | eqtri 2644 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘1), (𝑃‘2)} |
41 | 32, 13, 40 | 3eqtri 2648 | . . 3 ⊢ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} |
42 | 2wlklem 26563 | . . 3 ⊢ (∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) | |
43 | 27, 41, 42 | mpbir2an 955 | . 2 ⊢ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
44 | s2cli 13625 | . . . . . . 7 ⊢ 〈“00”〉 ∈ Word V | |
45 | 1, 44 | eqeltri 2697 | . . . . . 6 ⊢ 𝐹 ∈ Word V |
46 | wrddm 13312 | . . . . . 6 ⊢ (𝐹 ∈ Word V → dom 𝐹 = (0..^(#‘𝐹))) | |
47 | 45, 46 | ax-mp 5 | . . . . 5 ⊢ dom 𝐹 = (0..^(#‘𝐹)) |
48 | 47 | eqcomi 2631 | . . . 4 ⊢ (0..^(#‘𝐹)) = dom 𝐹 |
49 | 1 | dmeqi 5325 | . . . 4 ⊢ dom 𝐹 = dom 〈“00”〉 |
50 | s2dm 13635 | . . . 4 ⊢ dom 〈“00”〉 = {0, 1} | |
51 | 48, 49, 50 | 3eqtri 2648 | . . 3 ⊢ (0..^(#‘𝐹)) = {0, 1} |
52 | 51 | raleqi 3142 | . 2 ⊢ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
53 | 43, 52 | mpbir 221 | 1 ⊢ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 {cpr 4179 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 2c2 11070 ℤcz 11377 ..^cfzo 12465 #chash 13117 Word cword 13291 〈“cs1 13294 〈“cs2 13586 〈“cs3 13587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 |
This theorem is referenced by: wlk2v2e 27017 |
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