![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11246 | . 2 ⊢ 0 ∈ V | |
2 | 1ex 11248 | . 2 ⊢ 1 ∈ V | |
3 | 2fveq3 6907 | . . 3 ⊢ (𝑘 = 0 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘0))) | |
4 | fveq2 6902 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
5 | fv0p1e1 12373 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) | |
6 | 4, 5 | preq12d 4750 | . . 3 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)}) |
7 | 3, 6 | eqeq12d 2744 | . 2 ⊢ (𝑘 = 0 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})) |
8 | 2fveq3 6907 | . . 3 ⊢ (𝑘 = 1 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘1))) | |
9 | fveq2 6902 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1)) | |
10 | oveq1 7433 | . . . . . 6 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1)) | |
11 | 1p1e2 12375 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | eqtrdi 2784 | . . . . 5 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2) |
13 | 12 | fveq2d 6906 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2)) |
14 | 9, 13 | preq12d 4750 | . . 3 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)}) |
15 | 8, 14 | eqeq12d 2744 | . 2 ⊢ (𝑘 = 1 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
16 | 1, 2, 7, 15 | ralpr 4709 | 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 205 ∧ wa 394 = wceq 1533 ∀wral 3058 {cpr 4634 ‘cfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 + caddc 11149 2c2 12305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-2 12313 |
This theorem is referenced by: upgr2wlk 29502 usgr2wlkneq 29590 usgr2trlncl 29594 usgr2pthlem 29597 usgr2pth 29598 uspgrn2crct 29639 wlk2v2elem2 29986 |
Copyright terms: Public domain | W3C validator |