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Theorem finsumvtxdg2ssteplem4 28538
Description: Lemma for finsumvtxdg2sstep 28539. (Contributed by AV, 12-Dec-2021.)
Hypotheses
Ref Expression
finsumvtxdg2sstep.v 𝑉 = (Vtxβ€˜πΊ)
finsumvtxdg2sstep.e 𝐸 = (iEdgβ€˜πΊ)
finsumvtxdg2sstep.k 𝐾 = (𝑉 βˆ– {𝑁})
finsumvtxdg2sstep.i 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 βˆ‰ (πΈβ€˜π‘–)}
finsumvtxdg2sstep.p 𝑃 = (𝐸 β†Ύ 𝐼)
finsumvtxdg2sstep.s 𝑆 = ⟨𝐾, π‘ƒβŸ©
finsumvtxdg2ssteplem.j 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘–)}
Assertion
Ref Expression
finsumvtxdg2ssteplem4 ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ))) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = (2 Β· ((β™―β€˜π‘ƒ) + (β™―β€˜π½))))
Distinct variable groups:   𝑖,𝐸   𝑖,𝐺   𝑖,𝑁   𝑣,𝐸   𝑣,𝐺   𝑣,𝑁   𝑖,𝑉,𝑣   𝑖,𝐽   𝑣,𝐾
Allowed substitution hints:   𝑃(𝑣,𝑖)   𝑆(𝑣,𝑖)   𝐼(𝑣,𝑖)   𝐽(𝑣)   𝐾(𝑖)

Proof of Theorem finsumvtxdg2ssteplem4
StepHypRef Expression
1 finsumvtxdg2sstep.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
2 finsumvtxdg2sstep.e . . . . . . . 8 𝐸 = (iEdgβ€˜πΊ)
3 finsumvtxdg2sstep.k . . . . . . . 8 𝐾 = (𝑉 βˆ– {𝑁})
4 finsumvtxdg2sstep.i . . . . . . . 8 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 βˆ‰ (πΈβ€˜π‘–)}
5 finsumvtxdg2sstep.p . . . . . . . 8 𝑃 = (𝐸 β†Ύ 𝐼)
6 finsumvtxdg2sstep.s . . . . . . . 8 𝑆 = ⟨𝐾, π‘ƒβŸ©
7 finsumvtxdg2ssteplem.j . . . . . . . 8 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘–)}
81, 2, 3, 4, 5, 6, 7vtxdginducedm1fi 28534 . . . . . . 7 (𝐸 ∈ Fin β†’ βˆ€π‘£ ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) = (((VtxDegβ€˜π‘†)β€˜π‘£) + (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
98ad2antll 728 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ βˆ€π‘£ ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) = (((VtxDegβ€˜π‘†)β€˜π‘£) + (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
109sumeq2d 15594 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) = Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(((VtxDegβ€˜π‘†)β€˜π‘£) + (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
11 diffi 9130 . . . . . . . 8 (𝑉 ∈ Fin β†’ (𝑉 βˆ– {𝑁}) ∈ Fin)
1211adantr 482 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ (𝑉 βˆ– {𝑁}) ∈ Fin)
1312adantl 483 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (𝑉 βˆ– {𝑁}) ∈ Fin)
145dmeqi 5865 . . . . . . . . 9 dom 𝑃 = dom (𝐸 β†Ύ 𝐼)
15 finresfin 9221 . . . . . . . . . 10 (𝐸 ∈ Fin β†’ (𝐸 β†Ύ 𝐼) ∈ Fin)
16 dmfi 9281 . . . . . . . . . 10 ((𝐸 β†Ύ 𝐼) ∈ Fin β†’ dom (𝐸 β†Ύ 𝐼) ∈ Fin)
1715, 16syl 17 . . . . . . . . 9 (𝐸 ∈ Fin β†’ dom (𝐸 β†Ύ 𝐼) ∈ Fin)
1814, 17eqeltrid 2842 . . . . . . . 8 (𝐸 ∈ Fin β†’ dom 𝑃 ∈ Fin)
1918ad2antll 728 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ dom 𝑃 ∈ Fin)
203eqcomi 2746 . . . . . . . . 9 (𝑉 βˆ– {𝑁}) = 𝐾
2120eleq2i 2830 . . . . . . . 8 (𝑣 ∈ (𝑉 βˆ– {𝑁}) ↔ 𝑣 ∈ 𝐾)
2221biimpi 215 . . . . . . 7 (𝑣 ∈ (𝑉 βˆ– {𝑁}) β†’ 𝑣 ∈ 𝐾)
236fveq2i 6850 . . . . . . . . . 10 (Vtxβ€˜π‘†) = (Vtxβ€˜βŸ¨πΎ, π‘ƒβŸ©)
241fvexi 6861 . . . . . . . . . . . . 13 𝑉 ∈ V
2524difexi 5290 . . . . . . . . . . . 12 (𝑉 βˆ– {𝑁}) ∈ V
263, 25eqeltri 2834 . . . . . . . . . . 11 𝐾 ∈ V
272fvexi 6861 . . . . . . . . . . . . 13 𝐸 ∈ V
2827resex 5990 . . . . . . . . . . . 12 (𝐸 β†Ύ 𝐼) ∈ V
295, 28eqeltri 2834 . . . . . . . . . . 11 𝑃 ∈ V
3026, 29opvtxfvi 28002 . . . . . . . . . 10 (Vtxβ€˜βŸ¨πΎ, π‘ƒβŸ©) = 𝐾
3123, 30eqtr2i 2766 . . . . . . . . 9 𝐾 = (Vtxβ€˜π‘†)
321, 2, 3, 4, 5, 6vtxdginducedm1lem1 28529 . . . . . . . . . 10 (iEdgβ€˜π‘†) = 𝑃
3332eqcomi 2746 . . . . . . . . 9 𝑃 = (iEdgβ€˜π‘†)
34 eqid 2737 . . . . . . . . 9 dom 𝑃 = dom 𝑃
3531, 33, 34vtxdgfisnn0 28465 . . . . . . . 8 ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ 𝐾) β†’ ((VtxDegβ€˜π‘†)β€˜π‘£) ∈ β„•0)
3635nn0cnd 12482 . . . . . . 7 ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ 𝐾) β†’ ((VtxDegβ€˜π‘†)β€˜π‘£) ∈ β„‚)
3719, 22, 36syl2an 597 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ (𝑉 βˆ– {𝑁})) β†’ ((VtxDegβ€˜π‘†)β€˜π‘£) ∈ β„‚)
38 dmfi 9281 . . . . . . . . . . . 12 (𝐸 ∈ Fin β†’ dom 𝐸 ∈ Fin)
39 rabfi 9220 . . . . . . . . . . . 12 (dom 𝐸 ∈ Fin β†’ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘–)} ∈ Fin)
4038, 39syl 17 . . . . . . . . . . 11 (𝐸 ∈ Fin β†’ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘–)} ∈ Fin)
417, 40eqeltrid 2842 . . . . . . . . . 10 (𝐸 ∈ Fin β†’ 𝐽 ∈ Fin)
42 rabfi 9220 . . . . . . . . . 10 (𝐽 ∈ Fin β†’ {𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)} ∈ Fin)
43 hashcl 14263 . . . . . . . . . 10 ({𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)} ∈ Fin β†’ (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„•0)
4441, 42, 433syl 18 . . . . . . . . 9 (𝐸 ∈ Fin β†’ (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„•0)
4544nn0cnd 12482 . . . . . . . 8 (𝐸 ∈ Fin β†’ (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
4645ad2antll 728 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
4746adantr 482 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ (𝑉 βˆ– {𝑁})) β†’ (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
4813, 37, 47fsumadd 15632 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(((VtxDegβ€˜π‘†)β€˜π‘£) + (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})) = (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
4910, 48eqtrd 2777 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) = (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
503sumeq1i 15590 . . . . . 6 Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£)
5150eqeq1i 2742 . . . . 5 (Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ)) ↔ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ)))
52 oveq1 7369 . . . . 5 (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ)) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})) = ((2 Β· (β™―β€˜π‘ƒ)) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
5351, 52sylbi 216 . . . 4 (Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ)) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})) = ((2 Β· (β™―β€˜π‘ƒ)) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
5449, 53sylan9eq 2797 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ))) β†’ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) = ((2 Β· (β™―β€˜π‘ƒ)) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})))
5554oveq1d 7377 . 2 ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ))) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = (((2 Β· (β™―β€˜π‘ƒ)) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))))
5645adantl 483 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
5756adantr 482 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑣 ∈ (𝑉 βˆ– {𝑁})) β†’ (β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
5812, 57fsumcl 15625 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
59 hashcl 14263 . . . . . . . . . . 11 (𝐽 ∈ Fin β†’ (β™―β€˜π½) ∈ β„•0)
6041, 59syl 17 . . . . . . . . . 10 (𝐸 ∈ Fin β†’ (β™―β€˜π½) ∈ β„•0)
6160nn0cnd 12482 . . . . . . . . 9 (𝐸 ∈ Fin β†’ (β™―β€˜π½) ∈ β„‚)
6261adantl 483 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ (β™―β€˜π½) ∈ β„‚)
63 rabfi 9220 . . . . . . . . . . 11 (dom 𝐸 ∈ Fin β†’ {𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}} ∈ Fin)
64 hashcl 14263 . . . . . . . . . . 11 ({𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}} ∈ Fin β†’ (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}) ∈ β„•0)
6538, 63, 643syl 18 . . . . . . . . . 10 (𝐸 ∈ Fin β†’ (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}) ∈ β„•0)
6665nn0cnd 12482 . . . . . . . . 9 (𝐸 ∈ Fin β†’ (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}) ∈ β„‚)
6766adantl 483 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}) ∈ β„‚)
6858, 62, 67add12d 11388 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = ((β™―β€˜π½) + (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))))
6968adantl 483 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = ((β™―β€˜π½) + (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))))
701, 2, 3, 4, 5, 6, 7finsumvtxdg2ssteplem3 28537 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}})) = (β™―β€˜π½))
7170oveq2d 7378 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ ((β™―β€˜π½) + (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = ((β™―β€˜π½) + (β™―β€˜π½)))
72612timesd 12403 . . . . . . . 8 (𝐸 ∈ Fin β†’ (2 Β· (β™―β€˜π½)) = ((β™―β€˜π½) + (β™―β€˜π½)))
7372eqcomd 2743 . . . . . . 7 (𝐸 ∈ Fin β†’ ((β™―β€˜π½) + (β™―β€˜π½)) = (2 Β· (β™―β€˜π½)))
7473ad2antll 728 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ ((β™―β€˜π½) + (β™―β€˜π½)) = (2 Β· (β™―β€˜π½)))
7569, 71, 743eqtrd 2781 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = (2 Β· (β™―β€˜π½)))
7675oveq2d 7378 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ ((2 Β· (β™―β€˜π‘ƒ)) + (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}})))) = ((2 Β· (β™―β€˜π‘ƒ)) + (2 Β· (β™―β€˜π½))))
77 2cnd 12238 . . . . . . 7 (𝐸 ∈ Fin β†’ 2 ∈ β„‚)
785, 15eqeltrid 2842 . . . . . . . . 9 (𝐸 ∈ Fin β†’ 𝑃 ∈ Fin)
79 hashcl 14263 . . . . . . . . 9 (𝑃 ∈ Fin β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8078, 79syl 17 . . . . . . . 8 (𝐸 ∈ Fin β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8180nn0cnd 12482 . . . . . . 7 (𝐸 ∈ Fin β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
8277, 81mulcld 11182 . . . . . 6 (𝐸 ∈ Fin β†’ (2 Β· (β™―β€˜π‘ƒ)) ∈ β„‚)
8382ad2antll 728 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (2 Β· (β™―β€˜π‘ƒ)) ∈ β„‚)
8458adantl 483 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
8561, 66addcld 11181 . . . . . 6 (𝐸 ∈ Fin β†’ ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}})) ∈ β„‚)
8685ad2antll 728 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}})) ∈ β„‚)
8783, 84, 86addassd 11184 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (((2 Β· (β™―β€˜π‘ƒ)) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = ((2 Β· (β™―β€˜π‘ƒ)) + (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}})))))
88 2cnd 12238 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ 2 ∈ β„‚)
8981ad2antll 728 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
9061ad2antll 728 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (β™―β€˜π½) ∈ β„‚)
9188, 89, 90adddid 11186 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (2 Β· ((β™―β€˜π‘ƒ) + (β™―β€˜π½))) = ((2 Β· (β™―β€˜π‘ƒ)) + (2 Β· (β™―β€˜π½))))
9276, 87, 913eqtr4d 2787 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ (((2 Β· (β™―β€˜π‘ƒ)) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = (2 Β· ((β™―β€˜π‘ƒ) + (β™―β€˜π½))))
9392adantr 482 . 2 ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ))) β†’ (((2 Β· (β™―β€˜π‘ƒ)) + Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)})) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = (2 Β· ((β™―β€˜π‘ƒ) + (β™―β€˜π½))))
9455, 93eqtrd 2777 1 ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = (2 Β· (β™―β€˜π‘ƒ))) β†’ (Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜πΊ)β€˜π‘£) + ((β™―β€˜π½) + (β™―β€˜{𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}}))) = (2 Β· ((β™―β€˜π‘ƒ) + (β™―β€˜π½))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆ‰ wnel 3050  βˆ€wral 3065  {crab 3410  Vcvv 3448   βˆ– cdif 3912  {csn 4591  βŸ¨cop 4597  dom cdm 5638   β†Ύ cres 5640  β€˜cfv 6501  (class class class)co 7362  Fincfn 8890  β„‚cc 11056   + caddc 11061   Β· cmul 11063  2c2 12215  β„•0cn0 12420  β™―chash 14237  Ξ£csu 15577  Vtxcvtx 27989  iEdgciedg 27990  UPGraphcupgr 28073  VtxDegcvtxdg 28455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-rp 12923  df-xadd 13041  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578  df-vtx 27991  df-iedg 27992  df-edg 28041  df-uhgr 28051  df-upgr 28075  df-vtxdg 28456
This theorem is referenced by:  finsumvtxdg2sstep  28539
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