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Mirrors > Home > MPE Home > Th. List > wlk2v2elem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for wlk2v2e 27942: 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 6646 | . . . . . 6 ⊢ (𝐹‘0) = (〈“00”〉‘0) |
3 | 0z 11980 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
4 | s2fv0 14240 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘0) = 0) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘0) = 0 |
6 | 2, 5 | eqtri 2821 | . . . . 5 ⊢ (𝐹‘0) = 0 |
7 | 6 | fveq2i 6648 | . . . 4 ⊢ (𝐼‘(𝐹‘0)) = (𝐼‘0) |
8 | wlk2v2e.i | . . . . . 6 ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 | |
9 | 8 | fveq1i 6646 | . . . . 5 ⊢ (𝐼‘0) = (〈“{𝑋, 𝑌}”〉‘0) |
10 | prex 5298 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
11 | s1fv 13955 | . . . . . 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 2821 | . . . 4 ⊢ (𝐼‘0) = {𝑋, 𝑌} |
14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
15 | 14 | fveq1i 6646 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝑋𝑌𝑋”〉‘0) |
16 | wlk2v2e.x | . . . . . . . 8 ⊢ 𝑋 ∈ V | |
17 | s3fv0 14244 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘0) = 𝑋) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘0) = 𝑋 |
19 | 15, 18 | eqtri 2821 | . . . . . 6 ⊢ (𝑃‘0) = 𝑋 |
20 | 14 | fveq1i 6646 | . . . . . . 7 ⊢ (𝑃‘1) = (〈“𝑋𝑌𝑋”〉‘1) |
21 | wlk2v2e.y | . . . . . . . 8 ⊢ 𝑌 ∈ V | |
22 | s3fv1 14245 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (〈“𝑋𝑌𝑋”〉‘1) = 𝑌) | |
23 | 21, 22 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘1) = 𝑌 |
24 | 20, 23 | eqtri 2821 | . . . . . 6 ⊢ (𝑃‘1) = 𝑌 |
25 | 19, 24 | preq12i 4634 | . . . . 5 ⊢ {(𝑃‘0), (𝑃‘1)} = {𝑋, 𝑌} |
26 | 25 | eqcomi 2807 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘0), (𝑃‘1)} |
27 | 7, 13, 26 | 3eqtri 2825 | . . 3 ⊢ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} |
28 | 1 | fveq1i 6646 | . . . . . 6 ⊢ (𝐹‘1) = (〈“00”〉‘1) |
29 | s2fv1 14241 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘1) = 0) | |
30 | 3, 29 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘1) = 0 |
31 | 28, 30 | eqtri 2821 | . . . . 5 ⊢ (𝐹‘1) = 0 |
32 | 31 | fveq2i 6648 | . . . 4 ⊢ (𝐼‘(𝐹‘1)) = (𝐼‘0) |
33 | prcom 4628 | . . . . 5 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
34 | 14 | fveq1i 6646 | . . . . . . . 8 ⊢ (𝑃‘2) = (〈“𝑋𝑌𝑋”〉‘2) |
35 | s3fv2 14246 | . . . . . . . . 9 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘2) = 𝑋) | |
36 | 16, 35 | ax-mp 5 | . . . . . . . 8 ⊢ (〈“𝑋𝑌𝑋”〉‘2) = 𝑋 |
37 | 34, 36 | eqtri 2821 | . . . . . . 7 ⊢ (𝑃‘2) = 𝑋 |
38 | 24, 37 | preq12i 4634 | . . . . . 6 ⊢ {(𝑃‘1), (𝑃‘2)} = {𝑌, 𝑋} |
39 | 38 | eqcomi 2807 | . . . . 5 ⊢ {𝑌, 𝑋} = {(𝑃‘1), (𝑃‘2)} |
40 | 33, 39 | eqtri 2821 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘1), (𝑃‘2)} |
41 | 32, 13, 40 | 3eqtri 2825 | . . 3 ⊢ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} |
42 | 2wlklem 27457 | . . 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 710 | . 2 ⊢ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
44 | 14, 1 | 2wlkdlem2 27712 | . . 3 ⊢ (0..^(♯‘𝐹)) = {0, 1} |
45 | 44 | raleqi 3362 | . 2 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
46 | 43, 45 | mpbir 234 | 1 ⊢ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 {cpr 4527 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 2c2 11680 ℤcz 11969 ..^cfzo 13028 ♯chash 13686 〈“cs1 13940 〈“cs2 14194 〈“cs3 14195 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 |
This theorem is referenced by: wlk2v2e 27942 |
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