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Theorem vdgrun 26428
 Description: The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.)
Hypotheses
Ref Expression
vdgrun.e (𝜑𝐸 Fn 𝐴)
vdgrun.f (𝜑𝐹 Fn 𝐵)
vdgrun.a (𝜑𝐴𝑋)
vdgrun.b (𝜑𝐵𝑌)
vdgrun.i (𝜑 → (𝐴𝐵) = ∅)
vdgrun.ge (𝜑𝑉 UMGrph 𝐸)
vdgrun.gf (𝜑𝑉 UMGrph 𝐹)
vdgrun.u (𝜑𝑈𝑉)
Assertion
Ref Expression
vdgrun (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) +𝑒 ((𝑉 VDeg 𝐹)‘𝑈)))

Proof of Theorem vdgrun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elun 3715 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 727 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)))
3 andir 908 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
42, 3bitri 263 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
54abbii 2726 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
6 df-rab 2905 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))}
7 unab 3853 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
85, 6, 73eqtr4i 2642 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))})
9 df-rab 2905 . . . . . . . . 9 {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))}
10 vdgrun.e . . . . . . . . . . . . 13 (𝜑𝐸 Fn 𝐴)
1110adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐸 Fn 𝐴)
12 vdgrun.f . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐵)
1312adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐹 Fn 𝐵)
14 vdgrun.i . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = ∅)
1514adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐴𝐵) = ∅)
16 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
17 fvun1 6179 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐴)) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1811, 13, 15, 16, 17syl112anc 1322 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1918eleq2d 2673 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐸𝑥)))
2019rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
219, 20syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
22 df-rab 2905 . . . . . . . . 9 {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}
2310adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐸 Fn 𝐴)
2412adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹 Fn 𝐵)
2514adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → (𝐴𝐵) = ∅)
26 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝑥𝐵)
27 fvun2 6180 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐵)) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2823, 24, 25, 26, 27syl112anc 1322 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2928eleq2d 2673 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐹𝑥)))
3029rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3122, 30syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3221, 31uneq12d 3730 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
338, 32syl5eq 2656 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
3433fveq2d 6107 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
35 vdgrun.a . . . . . . 7 (𝜑𝐴𝑋)
36 rabexg 4739 . . . . . . 7 (𝐴𝑋 → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V)
3735, 36syl 17 . . . . . 6 (𝜑 → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V)
38 vdgrun.b . . . . . . 7 (𝜑𝐵𝑌)
39 rabexg 4739 . . . . . . 7 (𝐵𝑌 → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V)
4038, 39syl 17 . . . . . 6 (𝜑 → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V)
41 ssrab2 3650 . . . . . . . . 9 {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴
42 ssrab2 3650 . . . . . . . . 9 {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵
43 ss2in 3802 . . . . . . . . 9 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴 ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵) → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵))
4441, 42, 43mp2an 704 . . . . . . . 8 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵)
4544, 14syl5sseq 3616 . . . . . . 7 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅)
46 ss0 3926 . . . . . . 7 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅ → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
4745, 46syl 17 . . . . . 6 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
48 hashunx 13036 . . . . . 6 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V ∧ ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅) → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
4937, 40, 47, 48syl3anc 1318 . . . . 5 (𝜑 → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
5034, 49eqtrd 2644 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
511anbi1i 727 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))
52 andir 908 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5351, 52bitri 263 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5453abbii 2726 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
55 df-rab 2905 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
56 unab 3853 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
5754, 55, 563eqtr4i 2642 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})})
58 df-rab 2905 . . . . . . . . 9 {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
5918eqeq1d 2612 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐸𝑥) = {𝑈}))
6059rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
6158, 60syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
62 df-rab 2905 . . . . . . . . 9 {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
6328eqeq1d 2612 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐹𝑥) = {𝑈}))
6463rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6562, 64syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6661, 65uneq12d 3730 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6757, 66syl5eq 2656 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6867fveq2d 6107 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
69 rabexg 4739 . . . . . . 7 (𝐴𝑋 → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V)
7035, 69syl 17 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V)
71 rabexg 4739 . . . . . . 7 (𝐵𝑌 → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V)
7238, 71syl 17 . . . . . 6 (𝜑 → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V)
73 ssrab2 3650 . . . . . . . . 9 {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴
74 ssrab2 3650 . . . . . . . . 9 {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵
75 ss2in 3802 . . . . . . . . 9 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴 ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵) → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵))
7673, 74, 75mp2an 704 . . . . . . . 8 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵)
7776, 14syl5sseq 3616 . . . . . . 7 (𝜑 → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ ∅)
78 ss0 3926 . . . . . . 7 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ ∅ → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) = ∅)
7977, 78syl 17 . . . . . 6 (𝜑 → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) = ∅)
80 hashunx 13036 . . . . . 6 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V ∧ ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) = ∅) → (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8170, 72, 79, 80syl3anc 1318 . . . . 5 (𝜑 → (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8268, 81eqtrd 2644 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8350, 82oveq12d 6567 . . 3 (𝜑 → ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})) +𝑒 ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
84 hashxrcl 13010 . . . . . 6 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℝ*)
8537, 84syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℝ*)
86 hashnemnf 12994 . . . . . 6 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ≠ -∞)
8737, 86syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ≠ -∞)
8885, 87jca 553 . . . 4 (𝜑 → ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℝ* ∧ (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ≠ -∞))
89 hashxrcl 13010 . . . . . 6 ({𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℝ*)
9040, 89syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℝ*)
91 hashnemnf 12994 . . . . . 6 ({𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ≠ -∞)
9240, 91syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ≠ -∞)
9390, 92jca 553 . . . 4 (𝜑 → ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℝ* ∧ (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ≠ -∞))
94 hashxrcl 13010 . . . . . 6 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℝ*)
9570, 94syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℝ*)
96 hashnemnf 12994 . . . . . 6 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ≠ -∞)
9770, 96syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ≠ -∞)
9895, 97jca 553 . . . 4 (𝜑 → ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℝ* ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ≠ -∞))
99 hashxrcl 13010 . . . . . 6 ({𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℝ*)
10072, 99syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℝ*)
101 hashnemnf 12994 . . . . . 6 ({𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ≠ -∞)
10272, 101syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ≠ -∞)
103100, 102jca 553 . . . 4 (𝜑 → ((#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℝ* ∧ (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ≠ -∞))
10488, 93, 98, 103xadd4d 12005 . . 3 (𝜑 → (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})) +𝑒 ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) +𝑒 ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
10583, 104eqtrd 2644 . 2 (𝜑 → ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) +𝑒 ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
106 relumgra 25843 . . . 4 Rel UMGrph
107 vdgrun.ge . . . 4 (𝜑𝑉 UMGrph 𝐸)
108 brrelex 5080 . . . 4 ((Rel UMGrph ∧ 𝑉 UMGrph 𝐸) → 𝑉 ∈ V)
109106, 107, 108sylancr 694 . . 3 (𝜑𝑉 ∈ V)
110 fnun 5911 . . . 4 (((𝐸 Fn 𝐴𝐹 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐸𝐹) Fn (𝐴𝐵))
11110, 12, 14, 110syl21anc 1317 . . 3 (𝜑 → (𝐸𝐹) Fn (𝐴𝐵))
112 unexg 6857 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴𝐵) ∈ V)
11335, 38, 112syl2anc 691 . . 3 (𝜑 → (𝐴𝐵) ∈ V)
114 vdgrun.u . . 3 (𝜑𝑈𝑉)
115 vdgrval 26423 . . 3 (((𝑉 ∈ V ∧ (𝐸𝐹) Fn (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) ∧ 𝑈𝑉) → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})))
116109, 111, 113, 114, 115syl31anc 1321 . 2 (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})))
117 vdgrval 26423 . . . 4 (((𝑉 ∈ V ∧ 𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
118109, 10, 35, 114, 117syl31anc 1321 . . 3 (𝜑 → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
119 vdgrval 26423 . . . 4 (((𝑉 ∈ V ∧ 𝐹 Fn 𝐵𝐵𝑌) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
120109, 12, 38, 114, 119syl31anc 1321 . . 3 (𝜑 → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
121118, 120oveq12d 6567 . 2 (𝜑 → (((𝑉 VDeg 𝐸)‘𝑈) +𝑒 ((𝑉 VDeg 𝐹)‘𝑈)) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) +𝑒 ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
122105, 116, 1213eqtr4d 2654 1 (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) +𝑒 ((𝑉 VDeg 𝐹)‘𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   class class class wbr 4583  Rel wrel 5043   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  -∞cmnf 9951  ℝ*cxr 9952   +𝑒 cxad 11820  #chash 12979   UMGrph cumg 25841   VDeg cvdg 26420 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 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  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-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-hash 12980  df-umgra 25842  df-vdgr 26421 This theorem is referenced by: (None)
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