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Mirrors > Home > MPE Home > Th. List > wlk2v2elem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for wlk2v2e 30186: 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 6908 | . . . . . 6 ⊢ (𝐹‘0) = (〈“00”〉‘0) |
3 | 0z 12622 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
4 | s2fv0 14923 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘0) = 0) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘0) = 0 |
6 | 2, 5 | eqtri 2763 | . . . . 5 ⊢ (𝐹‘0) = 0 |
7 | 6 | fveq2i 6910 | . . . 4 ⊢ (𝐼‘(𝐹‘0)) = (𝐼‘0) |
8 | wlk2v2e.i | . . . . . 6 ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 | |
9 | 8 | fveq1i 6908 | . . . . 5 ⊢ (𝐼‘0) = (〈“{𝑋, 𝑌}”〉‘0) |
10 | prex 5443 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
11 | s1fv 14645 | . . . . . 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 2763 | . . . 4 ⊢ (𝐼‘0) = {𝑋, 𝑌} |
14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
15 | 14 | fveq1i 6908 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝑋𝑌𝑋”〉‘0) |
16 | wlk2v2e.x | . . . . . . . 8 ⊢ 𝑋 ∈ V | |
17 | s3fv0 14927 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘0) = 𝑋) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘0) = 𝑋 |
19 | 15, 18 | eqtri 2763 | . . . . . 6 ⊢ (𝑃‘0) = 𝑋 |
20 | 14 | fveq1i 6908 | . . . . . . 7 ⊢ (𝑃‘1) = (〈“𝑋𝑌𝑋”〉‘1) |
21 | wlk2v2e.y | . . . . . . . 8 ⊢ 𝑌 ∈ V | |
22 | s3fv1 14928 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (〈“𝑋𝑌𝑋”〉‘1) = 𝑌) | |
23 | 21, 22 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘1) = 𝑌 |
24 | 20, 23 | eqtri 2763 | . . . . . 6 ⊢ (𝑃‘1) = 𝑌 |
25 | 19, 24 | preq12i 4743 | . . . . 5 ⊢ {(𝑃‘0), (𝑃‘1)} = {𝑋, 𝑌} |
26 | 25 | eqcomi 2744 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘0), (𝑃‘1)} |
27 | 7, 13, 26 | 3eqtri 2767 | . . 3 ⊢ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} |
28 | 1 | fveq1i 6908 | . . . . . 6 ⊢ (𝐹‘1) = (〈“00”〉‘1) |
29 | s2fv1 14924 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘1) = 0) | |
30 | 3, 29 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘1) = 0 |
31 | 28, 30 | eqtri 2763 | . . . . 5 ⊢ (𝐹‘1) = 0 |
32 | 31 | fveq2i 6910 | . . . 4 ⊢ (𝐼‘(𝐹‘1)) = (𝐼‘0) |
33 | prcom 4737 | . . . . 5 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
34 | 14 | fveq1i 6908 | . . . . . . . 8 ⊢ (𝑃‘2) = (〈“𝑋𝑌𝑋”〉‘2) |
35 | s3fv2 14929 | . . . . . . . . 9 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘2) = 𝑋) | |
36 | 16, 35 | ax-mp 5 | . . . . . . . 8 ⊢ (〈“𝑋𝑌𝑋”〉‘2) = 𝑋 |
37 | 34, 36 | eqtri 2763 | . . . . . . 7 ⊢ (𝑃‘2) = 𝑋 |
38 | 24, 37 | preq12i 4743 | . . . . . 6 ⊢ {(𝑃‘1), (𝑃‘2)} = {𝑌, 𝑋} |
39 | 38 | eqcomi 2744 | . . . . 5 ⊢ {𝑌, 𝑋} = {(𝑃‘1), (𝑃‘2)} |
40 | 33, 39 | eqtri 2763 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘1), (𝑃‘2)} |
41 | 32, 13, 40 | 3eqtri 2767 | . . 3 ⊢ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} |
42 | 2wlklem 29700 | . . 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 711 | . 2 ⊢ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
44 | 14, 1 | 2wlkdlem2 29956 | . . 3 ⊢ (0..^(♯‘𝐹)) = {0, 1} |
45 | 44 | raleqi 3322 | . 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 231 | 1 ⊢ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 {cpr 4633 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 2c2 12319 ℤcz 12611 ..^cfzo 13691 ♯chash 14366 〈“cs1 14630 〈“cs2 14877 〈“cs3 14878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 |
This theorem is referenced by: wlk2v2e 30186 |
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