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Mirrors > Home > MPE Home > Th. List > 2clwwlk2 | Structured version Visualization version GIF version |
Description: The set (𝑋𝐶2) of double loops of length 2 on a vertex 𝑋 is equal to the set of closed walks with length 2 on 𝑋. Considered as "double loops", the first of the two closed walks/loops is degenerated, i.e., has length 0. (Contributed by AV, 18-Feb-2022.) (Revised by AV, 20-Apr-2022.) |
Ref | Expression |
---|---|
2clwwlk.c | ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
Ref | Expression |
---|---|
2clwwlk2 | ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶2) = (𝑋(ClWWalksNOn‘𝐺)2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12566 | . . . 4 ⊢ 2 ∈ ℤ | |
2 | uzid 12809 | . . . 4 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 2 ∈ (ℤ≥‘2) |
4 | 2clwwlk.c | . . . 4 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) | |
5 | 4 | 2clwwlk 29395 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 2 ∈ (ℤ≥‘2)) → (𝑋𝐶2) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)2) ∣ (𝑤‘(2 − 2)) = 𝑋}) |
6 | 3, 5 | mpan2 689 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶2) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)2) ∣ (𝑤‘(2 − 2)) = 𝑋}) |
7 | 2cn 12259 | . . . . . 6 ⊢ 2 ∈ ℂ | |
8 | 7 | subidi 11503 | . . . . 5 ⊢ (2 − 2) = 0 |
9 | 8 | fveq2i 6872 | . . . 4 ⊢ (𝑤‘(2 − 2)) = (𝑤‘0) |
10 | isclwwlknon 29139 | . . . . 5 ⊢ (𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)2) ↔ (𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋)) | |
11 | 10 | simprbi 497 | . . . 4 ⊢ (𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)2) → (𝑤‘0) = 𝑋) |
12 | 9, 11 | eqtrid 2783 | . . 3 ⊢ (𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)2) → (𝑤‘(2 − 2)) = 𝑋) |
13 | 12 | rabeqc 3437 | . 2 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)2) ∣ (𝑤‘(2 − 2)) = 𝑋} = (𝑋(ClWWalksNOn‘𝐺)2) |
14 | 6, 13 | eqtrdi 2787 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶2) = (𝑋(ClWWalksNOn‘𝐺)2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3425 ‘cfv 6523 (class class class)co 7384 ∈ cmpo 7386 0cc0 11082 − cmin 11416 2c2 12239 ℤcz 12530 ℤ≥cuz 12794 ClWWalksN cclwwlkn 29072 ClWWalksNOncclwwlknon 29135 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5269 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 ax-pre-mulgt0 11159 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-int 4935 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7340 df-ov 7387 df-oprab 7388 df-mpo 7389 df-om 7830 df-1st 7948 df-2nd 7949 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-1o 8439 df-oadd 8443 df-er 8677 df-map 8796 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-card 9906 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-sub 11418 df-neg 11419 df-nn 12185 df-2 12247 df-n0 12445 df-xnn0 12517 df-z 12531 df-uz 12795 df-fz 13457 df-fzo 13600 df-hash 14263 df-word 14437 df-clwwlk 29030 df-clwwlkn 29073 df-clwwlknon 29136 |
This theorem is referenced by: (None) |
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