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Mirrors > Home > MPE Home > Th. List > Mathboxes > eupth2lem3lem7 | Structured version Visualization version GIF version |
Description: Lemma for eupath2lem3 26506: Combining trlsegvdeg 41395, eupth2lem3lem3 41398, eupth2lem3lem4 41399 and eupth2lem3lem6 41401. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.) |
Ref | Expression |
---|---|
trlsegvdeg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
trlsegvdeg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
trlsegvdeg.f | ⊢ (𝜑 → Fun 𝐼) |
trlsegvdeg.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) |
trlsegvdeg.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
trlsegvdeg.w | ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) |
trlsegvdeg.vx | ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
trlsegvdeg.vy | ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) |
trlsegvdeg.vz | ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) |
trlsegvdeg.ix | ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
trlsegvdeg.iy | ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
trlsegvdeg.iz | ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
eupth2lem3.o | ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) |
eupth2lem3.e | ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
Ref | Expression |
---|---|
eupth2lem3lem7 | ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | trlsegvdeg.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | trlsegvdeg.f | . . . . 5 ⊢ (𝜑 → Fun 𝐼) | |
4 | trlsegvdeg.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) | |
5 | trlsegvdeg.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
6 | trlsegvdeg.w | . . . . 5 ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) | |
7 | trlsegvdeg.vx | . . . . 5 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) | |
8 | trlsegvdeg.vy | . . . . 5 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) | |
9 | trlsegvdeg.vz | . . . . 5 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
10 | trlsegvdeg.ix | . . . . 5 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
11 | trlsegvdeg.iy | . . . . 5 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
12 | trlsegvdeg.iz | . . . . 5 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | trlsegvdeg 41395 | . . . 4 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) |
14 | 13 | breq2d 4595 | . . 3 ⊢ (𝜑 → (2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))) |
15 | 14 | notbid 307 | . 2 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ ¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))) |
16 | eupth2lem3.o | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) | |
17 | eupth2lem3.e | . . . . 5 ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) | |
18 | ifpprsnss 40845 | . . . . 5 ⊢ ((𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 19 | eupth2lem3lem3 41398 | . . 3 ⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17 | eupth2lem3lem5 41400 | . . . . . . 7 ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ 𝒫 𝑉) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 19, 21 | eupth2lem3lem4 41399 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
23 | 22 | 3expa 1257 | . . . . 5 ⊢ (((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1))) ∧ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
24 | 23 | expcom 450 | . . . 4 ⊢ ((𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1))) → ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))) |
25 | neanior 2874 | . . . . 5 ⊢ ((𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1))) ↔ ¬ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) | |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17 | eupth2lem3lem6 41401 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
27 | 26 | 3expa 1257 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1))) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
28 | 27 | expcom 450 | . . . . 5 ⊢ ((𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1))) → ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))) |
29 | 25, 28 | sylbir 224 | . . . 4 ⊢ (¬ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1))) → ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))) |
30 | 24, 29 | pm2.61i 175 | . . 3 ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
31 | 20, 30 | pm2.61dane 2869 | . 2 ⊢ (𝜑 → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
32 | 15, 31 | bitrd 267 | 1 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 if-wif 1006 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {csn 4125 {cpr 4127 〈cop 4131 class class class wbr 4583 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 2c2 10947 ...cfz 12197 ..^cfzo 12334 #chash 12979 ∥ cdvds 14821 Vtxcvtx 25673 iEdgciedg 25674 VtxDegcvtxdg 40681 TrailSctrls 40899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-uhgr 25724 df-ushgr 25725 df-edga 25793 df-uspgr 40380 df-vtxdg 40682 df-1wlks 40800 df-trls 40901 |
This theorem is referenced by: eupth2lem3 41404 |
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