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Theorem vdgrun 26191
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 3711 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 726 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)))
3 andir 907 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
42, 3bitri 262 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
54abbii 2722 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
6 df-rab 2901 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))}
7 unab 3849 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
85, 6, 73eqtr4i 2638 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))})
9 df-rab 2901 . . . . . . . . 9 {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))}
10 vdgrun.e . . . . . . . . . . . . 13 (𝜑𝐸 Fn 𝐴)
1110adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐸 Fn 𝐴)
12 vdgrun.f . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐵)
1312adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐹 Fn 𝐵)
14 vdgrun.i . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = ∅)
1514adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐴𝐵) = ∅)
16 simpr 475 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
17 fvun1 6161 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐴)) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1811, 13, 15, 16, 17syl112anc 1321 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1918eleq2d 2669 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐸𝑥)))
2019rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
219, 20syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
22 df-rab 2901 . . . . . . . . 9 {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}
2310adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐸 Fn 𝐴)
2412adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹 Fn 𝐵)
2514adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → (𝐴𝐵) = ∅)
26 simpr 475 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝑥𝐵)
27 fvun2 6162 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐵)) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2823, 24, 25, 26, 27syl112anc 1321 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2928eleq2d 2669 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐹𝑥)))
3029rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3122, 30syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3221, 31uneq12d 3726 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
338, 32syl5eq 2652 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
3433fveq2d 6089 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
35 vdgrun.a . . . . . . 7 (𝜑𝐴𝑋)
36 rabexg 4731 . . . . . . 7 (𝐴𝑋 → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V)
3735, 36syl 17 . . . . . 6 (𝜑 → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V)
38 vdgrun.b . . . . . . 7 (𝜑𝐵𝑌)
39 rabexg 4731 . . . . . . 7 (𝐵𝑌 → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V)
4038, 39syl 17 . . . . . 6 (𝜑 → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V)
41 ssrab2 3646 . . . . . . . . 9 {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴
42 ssrab2 3646 . . . . . . . . 9 {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵
43 ss2in 3798 . . . . . . . . 9 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴 ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵) → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵))
4441, 42, 43mp2an 703 . . . . . . . 8 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵)
4544, 14syl5sseq 3612 . . . . . . 7 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅)
46 ss0 3922 . . . . . . 7 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅ → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
4745, 46syl 17 . . . . . 6 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
48 hashunx 12985 . . . . . 6 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V ∧ ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅) → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
4937, 40, 47, 48syl3anc 1317 . . . . 5 (𝜑 → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
5034, 49eqtrd 2640 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
511anbi1i 726 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))
52 andir 907 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5351, 52bitri 262 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5453abbii 2722 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
55 df-rab 2901 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
56 unab 3849 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
5754, 55, 563eqtr4i 2638 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})})
58 df-rab 2901 . . . . . . . . 9 {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
5918eqeq1d 2608 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐸𝑥) = {𝑈}))
6059rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
6158, 60syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
62 df-rab 2901 . . . . . . . . 9 {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
6328eqeq1d 2608 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐹𝑥) = {𝑈}))
6463rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6562, 64syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6661, 65uneq12d 3726 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6757, 66syl5eq 2652 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6867fveq2d 6089 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
69 rabexg 4731 . . . . . . 7 (𝐴𝑋 → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V)
7035, 69syl 17 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V)
71 rabexg 4731 . . . . . . 7 (𝐵𝑌 → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V)
7238, 71syl 17 . . . . . 6 (𝜑 → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V)
73 ssrab2 3646 . . . . . . . . 9 {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴
74 ssrab2 3646 . . . . . . . . 9 {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵
75 ss2in 3798 . . . . . . . . 9 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴 ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵) → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵))
7673, 74, 75mp2an 703 . . . . . . . 8 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵)
7776, 14syl5sseq 3612 . . . . . . 7 (𝜑 → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ ∅)
78 ss0 3922 . . . . . . 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 12985 . . . . . 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 1317 . . . . 5 (𝜑 → (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8268, 81eqtrd 2640 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8350, 82oveq12d 6542 . . 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 12959 . . . . . 6 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℝ*)
8537, 84syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℝ*)
86 hashnemnf 12943 . . . . . 6 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ V → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ≠ -∞)
8737, 86syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ≠ -∞)
8885, 87jca 552 . . . 4 (𝜑 → ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℝ* ∧ (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ≠ -∞))
89 hashxrcl 12959 . . . . . 6 ({𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℝ*)
9040, 89syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℝ*)
91 hashnemnf 12943 . . . . . 6 ({𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ V → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ≠ -∞)
9240, 91syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ≠ -∞)
9390, 92jca 552 . . . 4 (𝜑 → ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℝ* ∧ (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ≠ -∞))
94 hashxrcl 12959 . . . . . 6 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℝ*)
9570, 94syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℝ*)
96 hashnemnf 12943 . . . . . 6 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ≠ -∞)
9770, 96syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ≠ -∞)
9895, 97jca 552 . . . 4 (𝜑 → ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℝ* ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ≠ -∞))
99 hashxrcl 12959 . . . . . 6 ({𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℝ*)
10072, 99syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℝ*)
101 hashnemnf 12943 . . . . . 6 ({𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ V → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ≠ -∞)
10272, 101syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ≠ -∞)
103100, 102jca 552 . . . 4 (𝜑 → ((#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℝ* ∧ (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ≠ -∞))
10488, 93, 98, 103xadd4d 11959 . . 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 2640 . 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 25606 . . . 4 Rel UMGrph
107 vdgrun.ge . . . 4 (𝜑𝑉 UMGrph 𝐸)
108 brrelex 5067 . . . 4 ((Rel UMGrph ∧ 𝑉 UMGrph 𝐸) → 𝑉 ∈ V)
109106, 107, 108sylancr 693 . . 3 (𝜑𝑉 ∈ V)
110 fnun 5894 . . . 4 (((𝐸 Fn 𝐴𝐹 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐸𝐹) Fn (𝐴𝐵))
11110, 12, 14, 110syl21anc 1316 . . 3 (𝜑 → (𝐸𝐹) Fn (𝐴𝐵))
112 unexg 6831 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴𝐵) ∈ V)
11335, 38, 112syl2anc 690 . . 3 (𝜑 → (𝐴𝐵) ∈ V)
114 vdgrun.u . . 3 (𝜑𝑈𝑉)
115 vdgrval 26186 . . 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 1320 . 2 (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})))
117 vdgrval 26186 . . . 4 (((𝑉 ∈ V ∧ 𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
118109, 10, 35, 114, 117syl31anc 1320 . . 3 (𝜑 → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
119 vdgrval 26186 . . . 4 (((𝑉 ∈ V ∧ 𝐹 Fn 𝐵𝐵𝑌) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
120109, 12, 38, 114, 119syl31anc 1320 . . 3 (𝜑 → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) +𝑒 (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
121118, 120oveq12d 6542 . 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 2650 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 381  wa 382   = wceq 1474  wcel 1976  {cab 2592  wne 2776  {crab 2896  Vcvv 3169  cun 3534  cin 3535  wss 3536  c0 3870  {csn 4121   class class class wbr 4574  Rel wrel 5030   Fn wfn 5782  cfv 5787  (class class class)co 6524  -∞cmnf 9925  *cxr 9926   +𝑒 cxad 11773  #chash 12931   UMGrph cumg 25604   VDeg cvdg 26183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-n0 11137  df-z 11208  df-uz 11517  df-xadd 11776  df-hash 12932  df-umgra 25605  df-vdgr 26184
This theorem is referenced by: (None)
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