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Mirrors > Home > MPE Home > Th. List > finsumvtxdg2ssteplem1 | Structured version Visualization version GIF version |
Description: Lemma for finsumvtxdg2sstep 29350. (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 28902 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
2 | finsumvtxdg2sstep.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 2 | uhgrfun 28866 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸) |
5 | 4 | ad2antrr 725 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Fun 𝐸) |
6 | simprr 772 | . . 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 4076 | . . . 4 ⊢ 𝐼 ⊆ dom 𝐸 |
9 | 8 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝐼 ⊆ dom 𝐸) |
10 | hashreshashfun 14422 | . . 3 ⊢ ((Fun 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝐼 ⊆ dom 𝐸) → (♯‘𝐸) = ((♯‘(𝐸 ↾ 𝐼)) + (♯‘(dom 𝐸 ∖ 𝐼)))) | |
11 | 5, 6, 9, 10 | syl3anc 1369 | . 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 2736 | . . . . 5 ⊢ (𝐸 ↾ 𝐼) = 𝑃 |
14 | 13 | fveq2i 6894 | . . . 4 ⊢ (♯‘(𝐸 ↾ 𝐼)) = (♯‘𝑃) |
15 | 14 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘(𝐸 ↾ 𝐼)) = (♯‘𝑃)) |
16 | notrab 4307 | . . . . . 6 ⊢ (dom 𝐸 ∖ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}) = {𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ (𝐸‘𝑖)} | |
17 | 7 | difeq2i 4115 | . . . . . 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 3051 | . . . . . . . . 9 ⊢ (¬ 𝑁 ∉ (𝐸‘𝑖) ↔ 𝑁 ∈ (𝐸‘𝑖)) | |
20 | 19 | bicomi 223 | . . . . . . . 8 ⊢ (𝑁 ∈ (𝐸‘𝑖) ↔ ¬ 𝑁 ∉ (𝐸‘𝑖)) |
21 | 20 | rabbii 3433 | . . . . . . 7 ⊢ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} = {𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ (𝐸‘𝑖)} |
22 | 18, 21 | eqtri 2755 | . . . . . 6 ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ (𝐸‘𝑖)} |
23 | 16, 17, 22 | 3eqtr4i 2765 | . . . . 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 6895 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘(dom 𝐸 ∖ 𝐼)) = (♯‘𝐽)) |
26 | 15, 25 | oveq12d 7432 | . 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((♯‘(𝐸 ↾ 𝐼)) + (♯‘(dom 𝐸 ∖ 𝐼))) = ((♯‘𝑃) + (♯‘𝐽))) |
27 | 11, 26 | eqtrd 2767 | 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 395 = wceq 1534 ∈ wcel 2099 ∉ wnel 3041 {crab 3427 ∖ cdif 3941 ⊆ wss 3944 {csn 4624 ⟨cop 4630 dom cdm 5672 ↾ cres 5674 Fun wfun 6536 ‘cfv 6542 (class class class)co 7414 Fincfn 8955 + caddc 11133 ♯chash 14313 Vtxcvtx 28796 iEdgciedg 28797 UHGraphcuhgr 28856 UPGraphcupgr 28880 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-fz 13509 df-hash 14314 df-uhgr 28858 df-upgr 28882 |
This theorem is referenced by: finsumvtxdg2sstep 29350 |
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