![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wlkp1lem7 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 29615. (Contributed by AV, 6-Mar-2021.) |
Ref | Expression |
---|---|
wlkp1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wlkp1.i | ⊢ 𝐼 = (iEdg‘𝐺) |
wlkp1.f | ⊢ (𝜑 → Fun 𝐼) |
wlkp1.a | ⊢ (𝜑 → 𝐼 ∈ Fin) |
wlkp1.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
wlkp1.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
wlkp1.d | ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) |
wlkp1.w | ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
wlkp1.n | ⊢ 𝑁 = (♯‘𝐹) |
wlkp1.e | ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) |
wlkp1.x | ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) |
wlkp1.u | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) |
wlkp1.h | ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) |
wlkp1.q | ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) |
wlkp1.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
Ref | Expression |
---|---|
wlkp1lem7 | ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.x | . . 3 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) | |
2 | fveq2 6893 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑄‘𝑘) = (𝑄‘𝑁)) | |
3 | fveq2 6893 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
4 | 2, 3 | eqeq12d 2742 | . . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑄‘𝑘) = (𝑃‘𝑘) ↔ (𝑄‘𝑁) = (𝑃‘𝑁))) |
5 | wlkp1.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | wlkp1.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | wlkp1.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
8 | wlkp1.a | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
9 | wlkp1.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
10 | wlkp1.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
11 | wlkp1.d | . . . . . 6 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) | |
12 | wlkp1.w | . . . . . 6 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
13 | wlkp1.n | . . . . . 6 ⊢ 𝑁 = (♯‘𝐹) | |
14 | wlkp1.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) | |
15 | wlkp1.u | . . . . . 6 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) | |
16 | wlkp1.h | . . . . . 6 ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) | |
17 | wlkp1.q | . . . . . 6 ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) | |
18 | wlkp1.s | . . . . . 6 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
19 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 15, 16, 17, 18 | wlkp1lem5 29611 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) |
20 | wlkcl 29549 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
21 | 13 | eqcomi 2735 | . . . . . . . 8 ⊢ (♯‘𝐹) = 𝑁 |
22 | 21 | eleq1i 2817 | . . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0) |
23 | nn0fz0 13647 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) | |
24 | 22, 23 | sylbb 218 | . . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
25 | 12, 20, 24 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
26 | 4, 19, 25 | rspcdva 3608 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁)) |
27 | 17 | fveq1i 6894 | . . . . 5 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) |
28 | ovex 7449 | . . . . . 6 ⊢ (𝑁 + 1) ∈ V | |
29 | 5, 6, 7, 8, 9, 10, 11, 12, 13 | wlkp1lem1 29607 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃) |
30 | fsnunfv 7193 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) | |
31 | 28, 10, 29, 30 | mp3an2i 1463 | . . . . 5 ⊢ (𝜑 → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) |
32 | 27, 31 | eqtrid 2778 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶) |
33 | 26, 32 | preq12d 4740 | . . 3 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} = {(𝑃‘𝑁), 𝐶}) |
34 | fsnunfv 7193 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼) → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) | |
35 | 9, 14, 11, 34 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) |
36 | 1, 33, 35 | 3sstr4d 4026 | . 2 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
37 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 15, 16 | wlkp1lem3 29609 | . 2 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
38 | 36, 37 | sseqtrrd 4020 | 1 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∪ cun 3944 ⊆ wss 3946 {csn 4623 {cpr 4625 〈cop 4629 class class class wbr 5145 dom cdm 5674 Fun wfun 6540 ‘cfv 6546 (class class class)co 7416 Fincfn 8966 0cc0 11149 1c1 11150 + caddc 11152 ℕ0cn0 12518 ...cfz 13532 ♯chash 14342 Vtxcvtx 28929 iEdgciedg 28930 Edgcedg 28980 Walkscwlks 29530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-hash 14343 df-word 14518 df-wlks 29533 |
This theorem is referenced by: wlkp1lem8 29614 |
Copyright terms: Public domain | W3C validator |