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Mirrors > Home > MPE Home > Th. List > finsumvtxdg2ssteplem1 | Structured version Visualization version GIF version |
Description: Lemma for finsumvtxdg2sstep 27325. (Contributed by AV, 15-Dec-2021.) |
Ref | Expression |
---|---|
finsumvtxdg2sstep.v | ⊢ 𝑉 = (Vtx‘𝐺) |
finsumvtxdg2sstep.e | ⊢ 𝐸 = (iEdg‘𝐺) |
finsumvtxdg2sstep.k | ⊢ 𝐾 = (𝑉 ∖ {𝑁}) |
finsumvtxdg2sstep.i | ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} |
finsumvtxdg2sstep.p | ⊢ 𝑃 = (𝐸 ↾ 𝐼) |
finsumvtxdg2sstep.s | ⊢ 𝑆 = 〈𝐾, 𝑃〉 |
finsumvtxdg2ssteplem.j | ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} |
Ref | Expression |
---|---|
finsumvtxdg2ssteplem1 | ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgruhgr 26881 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
2 | finsumvtxdg2sstep.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 2 | uhgrfun 26845 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸) |
5 | 4 | ad2antrr 724 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Fun 𝐸) |
6 | simprr 771 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝐸 ∈ Fin) | |
7 | finsumvtxdg2sstep.i | . . . . 5 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} | |
8 | 7 | ssrab3 4057 | . . . 4 ⊢ 𝐼 ⊆ dom 𝐸 |
9 | 8 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝐼 ⊆ dom 𝐸) |
10 | hashreshashfun 13794 | . . 3 ⊢ ((Fun 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝐼 ⊆ dom 𝐸) → (♯‘𝐸) = ((♯‘(𝐸 ↾ 𝐼)) + (♯‘(dom 𝐸 ∖ 𝐼)))) | |
11 | 5, 6, 9, 10 | syl3anc 1367 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘(𝐸 ↾ 𝐼)) + (♯‘(dom 𝐸 ∖ 𝐼)))) |
12 | finsumvtxdg2sstep.p | . . . . . 6 ⊢ 𝑃 = (𝐸 ↾ 𝐼) | |
13 | 12 | eqcomi 2830 | . . . . 5 ⊢ (𝐸 ↾ 𝐼) = 𝑃 |
14 | 13 | fveq2i 6668 | . . . 4 ⊢ (♯‘(𝐸 ↾ 𝐼)) = (♯‘𝑃) |
15 | 14 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘(𝐸 ↾ 𝐼)) = (♯‘𝑃)) |
16 | notrab 4280 | . . . . . 6 ⊢ (dom 𝐸 ∖ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}) = {𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ (𝐸‘𝑖)} | |
17 | 7 | difeq2i 4096 | . . . . . 6 ⊢ (dom 𝐸 ∖ 𝐼) = (dom 𝐸 ∖ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}) |
18 | finsumvtxdg2ssteplem.j | . . . . . . 7 ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} | |
19 | nnel 3132 | . . . . . . . . 9 ⊢ (¬ 𝑁 ∉ (𝐸‘𝑖) ↔ 𝑁 ∈ (𝐸‘𝑖)) | |
20 | 19 | bicomi 226 | . . . . . . . 8 ⊢ (𝑁 ∈ (𝐸‘𝑖) ↔ ¬ 𝑁 ∉ (𝐸‘𝑖)) |
21 | 20 | rabbii 3474 | . . . . . . 7 ⊢ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} = {𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ (𝐸‘𝑖)} |
22 | 18, 21 | eqtri 2844 | . . . . . 6 ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ (𝐸‘𝑖)} |
23 | 16, 17, 22 | 3eqtr4i 2854 | . . . . 5 ⊢ (dom 𝐸 ∖ 𝐼) = 𝐽 |
24 | 23 | a1i 11 | . . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (dom 𝐸 ∖ 𝐼) = 𝐽) |
25 | 24 | fveq2d 6669 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘(dom 𝐸 ∖ 𝐼)) = (♯‘𝐽)) |
26 | 15, 25 | oveq12d 7168 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((♯‘(𝐸 ↾ 𝐼)) + (♯‘(dom 𝐸 ∖ 𝐼))) = ((♯‘𝑃) + (♯‘𝐽))) |
27 | 11, 26 | eqtrd 2856 | 1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 {crab 3142 ∖ cdif 3933 ⊆ wss 3936 {csn 4561 〈cop 4567 dom cdm 5550 ↾ cres 5552 Fun wfun 6344 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 + caddc 10534 ♯chash 13684 Vtxcvtx 26775 iEdgciedg 26776 UHGraphcuhgr 26835 UPGraphcupgr 26859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 df-uhgr 26837 df-upgr 26861 |
This theorem is referenced by: finsumvtxdg2sstep 27325 |
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