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Theorem finsumvtxdg2ssteplem4 28805
Description: Lemma for finsumvtxdg2sstep 28806. (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 28801 . . . . . . 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 15648 . . . . 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 9179 . . . . . . . 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 5905 . . . . . . . . 9 dom 𝑃 = dom (𝐸 β†Ύ 𝐼)
15 finresfin 9270 . . . . . . . . . 10 (𝐸 ∈ Fin β†’ (𝐸 β†Ύ 𝐼) ∈ Fin)
16 dmfi 9330 . . . . . . . . . 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 2838 . . . . . . . 8 (𝐸 ∈ Fin β†’ dom 𝑃 ∈ Fin)
1918ad2antll 728 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) β†’ dom 𝑃 ∈ Fin)
203eqcomi 2742 . . . . . . . . 9 (𝑉 βˆ– {𝑁}) = 𝐾
2120eleq2i 2826 . . . . . . . 8 (𝑣 ∈ (𝑉 βˆ– {𝑁}) ↔ 𝑣 ∈ 𝐾)
2221biimpi 215 . . . . . . 7 (𝑣 ∈ (𝑉 βˆ– {𝑁}) β†’ 𝑣 ∈ 𝐾)
236fveq2i 6895 . . . . . . . . . 10 (Vtxβ€˜π‘†) = (Vtxβ€˜βŸ¨πΎ, π‘ƒβŸ©)
241fvexi 6906 . . . . . . . . . . . . 13 𝑉 ∈ V
2524difexi 5329 . . . . . . . . . . . 12 (𝑉 βˆ– {𝑁}) ∈ V
263, 25eqeltri 2830 . . . . . . . . . . 11 𝐾 ∈ V
272fvexi 6906 . . . . . . . . . . . . 13 𝐸 ∈ V
2827resex 6030 . . . . . . . . . . . 12 (𝐸 β†Ύ 𝐼) ∈ V
295, 28eqeltri 2830 . . . . . . . . . . 11 𝑃 ∈ V
3026, 29opvtxfvi 28269 . . . . . . . . . 10 (Vtxβ€˜βŸ¨πΎ, π‘ƒβŸ©) = 𝐾
3123, 30eqtr2i 2762 . . . . . . . . 9 𝐾 = (Vtxβ€˜π‘†)
321, 2, 3, 4, 5, 6vtxdginducedm1lem1 28796 . . . . . . . . . 10 (iEdgβ€˜π‘†) = 𝑃
3332eqcomi 2742 . . . . . . . . 9 𝑃 = (iEdgβ€˜π‘†)
34 eqid 2733 . . . . . . . . 9 dom 𝑃 = dom 𝑃
3531, 33, 34vtxdgfisnn0 28732 . . . . . . . 8 ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ 𝐾) β†’ ((VtxDegβ€˜π‘†)β€˜π‘£) ∈ β„•0)
3635nn0cnd 12534 . . . . . . 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 9330 . . . . . . . . . . . 12 (𝐸 ∈ Fin β†’ dom 𝐸 ∈ Fin)
39 rabfi 9269 . . . . . . . . . . . 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 2838 . . . . . . . . . 10 (𝐸 ∈ Fin β†’ 𝐽 ∈ Fin)
42 rabfi 9269 . . . . . . . . . 10 (𝐽 ∈ Fin β†’ {𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)} ∈ Fin)
43 hashcl 14316 . . . . . . . . . 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 12534 . . . . . . . 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 15686 . . . . 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 2773 . . . 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 15644 . . . . . 6 Σ𝑣 ∈ 𝐾 ((VtxDegβ€˜π‘†)β€˜π‘£) = Σ𝑣 ∈ (𝑉 βˆ– {𝑁})((VtxDegβ€˜π‘†)β€˜π‘£)
5150eqeq1i 2738 . . . . 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 7416 . . . . 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 2793 . . 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 7424 . 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 15679 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ Σ𝑣 ∈ (𝑉 βˆ– {𝑁})(β™―β€˜{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (πΈβ€˜π‘–)}) ∈ β„‚)
59 hashcl 14316 . . . . . . . . . . 11 (𝐽 ∈ Fin β†’ (β™―β€˜π½) ∈ β„•0)
6041, 59syl 17 . . . . . . . . . 10 (𝐸 ∈ Fin β†’ (β™―β€˜π½) ∈ β„•0)
6160nn0cnd 12534 . . . . . . . . 9 (𝐸 ∈ Fin β†’ (β™―β€˜π½) ∈ β„‚)
6261adantl 483 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) β†’ (β™―β€˜π½) ∈ β„‚)
63 rabfi 9269 . . . . . . . . . . 11 (dom 𝐸 ∈ Fin β†’ {𝑖 ∈ dom 𝐸 ∣ (πΈβ€˜π‘–) = {𝑁}} ∈ Fin)
64 hashcl 14316 . . . . . . . . . . 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 12534 . . . . . . . . 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 11440 . . . . . . 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 28804 . . . . . . 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 7425 . . . . . 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 12455 . . . . . . . 8 (𝐸 ∈ Fin β†’ (2 Β· (β™―β€˜π½)) = ((β™―β€˜π½) + (β™―β€˜π½)))
7372eqcomd 2739 . . . . . . 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 2777 . . . . 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 7425 . . . 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 12290 . . . . . . 7 (𝐸 ∈ Fin β†’ 2 ∈ β„‚)
785, 15eqeltrid 2838 . . . . . . . . 9 (𝐸 ∈ Fin β†’ 𝑃 ∈ Fin)
79 hashcl 14316 . . . . . . . . 9 (𝑃 ∈ Fin β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8078, 79syl 17 . . . . . . . 8 (𝐸 ∈ Fin β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8180nn0cnd 12534 . . . . . . 7 (𝐸 ∈ Fin β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
8277, 81mulcld 11234 . . . . . 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 11233 . . . . . 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 11236 . . . 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 12290 . . . . 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 11238 . . . 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 2783 . . 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 2773 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 3047  βˆ€wral 3062  {crab 3433  Vcvv 3475   βˆ– cdif 3946  {csn 4629  βŸ¨cop 4635  dom cdm 5677   β†Ύ cres 5679  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  β„‚cc 11108   + caddc 11113   Β· cmul 11115  2c2 12267  β„•0cn0 12472  β™―chash 14290  Ξ£csu 15632  Vtxcvtx 28256  iEdgciedg 28257  UPGraphcupgr 28340  VtxDegcvtxdg 28722
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-oi 9505  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-rp 12975  df-xadd 13093  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-vtx 28258  df-iedg 28259  df-edg 28308  df-uhgr 28318  df-upgr 28342  df-vtxdg 28723
This theorem is referenced by:  finsumvtxdg2sstep  28806
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