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Mirrors > Home > MPE Home > Th. List > 2wlklem | Structured version Visualization version GIF version |
Description: Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Ref | Expression |
---|---|
2wlklem | ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10470 | . 2 ⊢ 0 ∈ V | |
2 | 1ex 10472 | . 2 ⊢ 1 ∈ V | |
3 | 2fveq3 6535 | . . 3 ⊢ (𝑘 = 0 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘0))) | |
4 | fveq2 6530 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
5 | fv0p1e1 11597 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) | |
6 | 4, 5 | preq12d 4578 | . . 3 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)}) |
7 | 3, 6 | eqeq12d 2808 | . 2 ⊢ (𝑘 = 0 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})) |
8 | 2fveq3 6535 | . . 3 ⊢ (𝑘 = 1 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘1))) | |
9 | fveq2 6530 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1)) | |
10 | oveq1 7014 | . . . . . 6 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1)) | |
11 | 1p1e2 11599 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | syl6eq 2845 | . . . . 5 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2) |
13 | 12 | fveq2d 6534 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2)) |
14 | 9, 13 | preq12d 4578 | . . 3 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)}) |
15 | 8, 14 | eqeq12d 2808 | . 2 ⊢ (𝑘 = 1 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
16 | 1, 2, 7, 15 | ralpr 4537 | 1 ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1520 ∀wral 3103 {cpr 4468 ‘cfv 6217 (class class class)co 7007 0cc0 10372 1c1 10373 + caddc 10375 2c2 11529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-ov 7010 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-ltxr 10515 df-2 11537 |
This theorem is referenced by: upgr2wlk 27120 usgr2wlkneq 27212 usgr2trlncl 27216 usgr2pthlem 27219 usgr2pth 27220 uspgrn2crct 27261 wlk2v2elem2 27610 |
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