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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 10321 | . 2 ⊢ 0 ∈ V | |
2 | 1ex 10323 | . 2 ⊢ 1 ∈ V | |
3 | 2fveq3 6415 | . . 3 ⊢ (𝑘 = 0 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘0))) | |
4 | fveq2 6410 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
5 | fv0p1e1 11440 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) | |
6 | 4, 5 | preq12d 4464 | . . 3 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)}) |
7 | 3, 6 | eqeq12d 2813 | . 2 ⊢ (𝑘 = 0 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})) |
8 | 2fveq3 6415 | . . 3 ⊢ (𝑘 = 1 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘1))) | |
9 | fveq2 6410 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1)) | |
10 | oveq1 6884 | . . . . . 6 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1)) | |
11 | 1p1e2 11442 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | syl6eq 2848 | . . . . 5 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2) |
13 | 12 | fveq2d 6414 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2)) |
14 | 9, 13 | preq12d 4464 | . . 3 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)}) |
15 | 8, 14 | eqeq12d 2813 | . 2 ⊢ (𝑘 = 1 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
16 | 1, 2, 7, 15 | ralpr 4427 | 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 198 ∧ wa 385 = wceq 1653 ∀wral 3088 {cpr 4369 ‘cfv 6100 (class class class)co 6877 0cc0 10223 1c1 10224 + caddc 10226 2c2 11365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-op 4374 df-uni 4628 df-br 4843 df-opab 4905 df-mpt 4922 df-id 5219 df-po 5232 df-so 5233 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-ov 6880 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-pnf 10364 df-mnf 10365 df-ltxr 10367 df-2 11373 |
This theorem is referenced by: upgr2wlk 26910 usgr2wlkneq 27003 usgr2trlncl 27007 usgr2pthlem 27010 usgr2pth 27011 uspgrn2crct 27052 wlk2v2elem2 27493 |
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