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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1wlk2v2elem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 1wlk2v2e 41324: 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 |
---|---|
1wlk2v2e.i | ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 |
1wlk2v2e.f | ⊢ 𝐹 = 〈“00”〉 |
1wlk2v2e.x | ⊢ 𝑋 ∈ V |
1wlk2v2e.y | ⊢ 𝑌 ∈ V |
1wlk2v2e.p | ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 |
Ref | Expression |
---|---|
1wlk2v2elem2 | ⊢ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
2 | 1 | fveq1i 6104 | . . . . . 6 ⊢ (𝐹‘0) = (〈“00”〉‘0) |
3 | 0z 11265 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
4 | s2fv0 13482 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘0) = 0) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘0) = 0 |
6 | 2, 5 | eqtri 2632 | . . . . 5 ⊢ (𝐹‘0) = 0 |
7 | 6 | fveq2i 6106 | . . . 4 ⊢ (𝐼‘(𝐹‘0)) = (𝐼‘0) |
8 | 1wlk2v2e.i | . . . . . 6 ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 | |
9 | 8 | fveq1i 6104 | . . . . 5 ⊢ (𝐼‘0) = (〈“{𝑋, 𝑌}”〉‘0) |
10 | prex 4836 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
11 | s1fv 13243 | . . . . . 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 2632 | . . . 4 ⊢ (𝐼‘0) = {𝑋, 𝑌} |
14 | 1wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
15 | 14 | fveq1i 6104 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝑋𝑌𝑋”〉‘0) |
16 | 1wlk2v2e.x | . . . . . . . 8 ⊢ 𝑋 ∈ V | |
17 | s3fv0 13486 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘0) = 𝑋) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘0) = 𝑋 |
19 | 15, 18 | eqtri 2632 | . . . . . 6 ⊢ (𝑃‘0) = 𝑋 |
20 | 14 | fveq1i 6104 | . . . . . . 7 ⊢ (𝑃‘1) = (〈“𝑋𝑌𝑋”〉‘1) |
21 | 1wlk2v2e.y | . . . . . . . 8 ⊢ 𝑌 ∈ V | |
22 | s3fv1 13487 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (〈“𝑋𝑌𝑋”〉‘1) = 𝑌) | |
23 | 21, 22 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘1) = 𝑌 |
24 | 20, 23 | eqtri 2632 | . . . . . 6 ⊢ (𝑃‘1) = 𝑌 |
25 | 19, 24 | preq12i 4217 | . . . . 5 ⊢ {(𝑃‘0), (𝑃‘1)} = {𝑋, 𝑌} |
26 | 25 | eqcomi 2619 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘0), (𝑃‘1)} |
27 | 7, 13, 26 | 3eqtri 2636 | . . 3 ⊢ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} |
28 | 1 | fveq1i 6104 | . . . . . 6 ⊢ (𝐹‘1) = (〈“00”〉‘1) |
29 | s2fv1 13483 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘1) = 0) | |
30 | 3, 29 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘1) = 0 |
31 | 28, 30 | eqtri 2632 | . . . . 5 ⊢ (𝐹‘1) = 0 |
32 | 31 | fveq2i 6106 | . . . 4 ⊢ (𝐼‘(𝐹‘1)) = (𝐼‘0) |
33 | prcom 4211 | . . . . 5 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
34 | 14 | fveq1i 6104 | . . . . . . . 8 ⊢ (𝑃‘2) = (〈“𝑋𝑌𝑋”〉‘2) |
35 | s3fv2 13488 | . . . . . . . . 9 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘2) = 𝑋) | |
36 | 16, 35 | ax-mp 5 | . . . . . . . 8 ⊢ (〈“𝑋𝑌𝑋”〉‘2) = 𝑋 |
37 | 34, 36 | eqtri 2632 | . . . . . . 7 ⊢ (𝑃‘2) = 𝑋 |
38 | 24, 37 | preq12i 4217 | . . . . . 6 ⊢ {(𝑃‘1), (𝑃‘2)} = {𝑌, 𝑋} |
39 | 38 | eqcomi 2619 | . . . . 5 ⊢ {𝑌, 𝑋} = {(𝑃‘1), (𝑃‘2)} |
40 | 33, 39 | eqtri 2632 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘1), (𝑃‘2)} |
41 | 32, 13, 40 | 3eqtri 2636 | . . 3 ⊢ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} |
42 | 2wlklem 26094 | . . 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 957 | . 2 ⊢ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
44 | s2cli 13475 | . . . . . . 7 ⊢ 〈“00”〉 ∈ Word V | |
45 | 1, 44 | eqeltri 2684 | . . . . . 6 ⊢ 𝐹 ∈ Word V |
46 | wrddm 13167 | . . . . . 6 ⊢ (𝐹 ∈ Word V → dom 𝐹 = (0..^(#‘𝐹))) | |
47 | 45, 46 | ax-mp 5 | . . . . 5 ⊢ dom 𝐹 = (0..^(#‘𝐹)) |
48 | 47 | eqcomi 2619 | . . . 4 ⊢ (0..^(#‘𝐹)) = dom 𝐹 |
49 | 1 | dmeqi 5247 | . . . 4 ⊢ dom 𝐹 = dom 〈“00”〉 |
50 | s2dm 13485 | . . . 4 ⊢ dom 〈“00”〉 = {0, 1} | |
51 | 48, 49, 50 | 3eqtri 2636 | . . 3 ⊢ (0..^(#‘𝐹)) = {0, 1} |
52 | 51 | raleqi 3119 | . 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 220 | 1 ⊢ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 {cpr 4127 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 2c2 10947 ℤcz 11254 ..^cfzo 12334 #chash 12979 Word cword 13146 〈“cs1 13149 〈“cs2 13437 〈“cs3 13438 |
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-2 10956 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 df-s2 13444 df-s3 13445 |
This theorem is referenced by: 1wlk2v2e 41324 |
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