Theorem vdiscusgr

Description: In a finite complete simple graph with n vertices every vertex has degree n - 1 . (Contributed by Alexander van der Vekens, 14-Jul-2018) (Revised by AV, 17-Dec-2020)

	Ref	Expression
Hypothesis	hashnbusgrvd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	vdiscusgr	⊢ ( 𝐺 ∈ FinUSGraph → ( ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) → 𝐺 ∈ ComplUSGraph ) )

Proof

Step	Hyp	Ref	Expression
1		hashnbusgrvd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	uvtxisvtx	⊢ ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) → 𝑛 ∈ 𝑉 )
3		fveqeq2	⊢ ( 𝑣 = 𝑛 → ( ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )
4	3	rspccv	⊢ ( ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) → ( 𝑛 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )
5	4	adantl	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝑛 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )
6	5	imp	⊢ ( ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ∧ 𝑛 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )
7	1	usgruvtxvdb	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑛 ∈ 𝑉 ) → ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )
8	7	adantlr	⊢ ( ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ∧ 𝑛 ∈ 𝑉 ) → ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )
9	6 8	mpbird	⊢ ( ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ∧ 𝑛 ∈ 𝑉 ) → 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) )
10	9	ex	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝑛 ∈ 𝑉 → 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ) )
11	2 10	impbid2	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ↔ 𝑛 ∈ 𝑉 ) )
12	11	eqrdv	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( UnivVtx ‘ 𝐺 ) = 𝑉 )
13		fusgrusgr	⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
14	1	cusgruvtxb	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) )
15	13 14	syl	⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) )
16	15	adantr	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) )
17	12 16	mpbird	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → 𝐺 ∈ ComplUSGraph )
18	17	ex	⊢ ( 𝐺 ∈ FinUSGraph → ( ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) → 𝐺 ∈ ComplUSGraph ) )