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Theorem sumodd 15729

Description: If every term in a sum is odd, then the sum is even iff the number of terms in the sum is even. (Contributed by AV, 14-Aug-2021.)

**Hypotheses**
Ref	Expression
sumeven.a	⊢ (𝜑 → 𝐴 ∈ Fin)
sumeven.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)
sumodd.o	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵)

**Assertion**
Ref	Expression
sumodd	⊢ (𝜑 → (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵))

Distinct variable groups: 𝐴,𝑘 𝜑,𝑘 Allowed substitution hint: 𝐵(𝑘)

Proof of Theorem sumodd Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6645	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
2		hash0 13724	. . . . 5 ⊢ (♯‘∅) = 0
3	1, 2	eqtrdi 2849	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
4	3	breq2d 5042	. . 3 ⊢ (𝑥 = ∅ → (2 ∥ (♯‘𝑥) ↔ 2 ∥ 0))
5		sumeq1 15037	. . . . 5 ⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
6		sum0 15070	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
7	5, 6	eqtrdi 2849	. . . 4 ⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 𝐵 = 0)
8	7	breq2d 5042	. . 3 ⊢ (𝑥 = ∅ → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ 0))
9	4, 8	bibi12d 349	. 2 ⊢ (𝑥 = ∅ → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ 0 ↔ 2 ∥ 0)))
10		fveq2 6645	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
11	10	breq2d 5042	. . 3 ⊢ (𝑥 = 𝑦 → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘𝑦)))
12		sumeq1 15037	. . . 4 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
13	12	breq2d 5042	. . 3 ⊢ (𝑥 = 𝑦 → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
14	11, 13	bibi12d 349	. 2 ⊢ (𝑥 = 𝑦 → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ (♯‘𝑦) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵)))
15		fveq2 6645	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	breq2d 5042	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘(𝑦 ∪ {𝑧}))))
17		sumeq1 15037	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
18	17	breq2d 5042	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
19	16, 18	bibi12d 349	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
20		fveq2 6645	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
21	20	breq2d 5042	. . 3 ⊢ (𝑥 = 𝐴 → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘𝐴)))
22		sumeq1 15037	. . . 4 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
23	22	breq2d 5042	. . 3 ⊢ (𝑥 = 𝐴 → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵))
24	21, 23	bibi12d 349	. 2 ⊢ (𝑥 = 𝐴 → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵)))
25		biidd 265	. 2 ⊢ (𝜑 → (2 ∥ 0 ↔ 2 ∥ 0))
26		eldifi 4054	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → 𝑧 ∈ 𝐴)
27	26	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑧 ∈ 𝐴)
28	27	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
29		sumeven.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)
30	29	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)
31	30	ralrimiva 3149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ)
32		rspcsbela 4343	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
33	28, 31, 32	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
34		sumodd.o	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵)
35	34	ralrimiva 3149	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ¬ 2 ∥ 𝐵)
36		nfcv 2955	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘2
37		nfcv 2955	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ∥
38		nfcsb1v 3852	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
39	36, 37, 38	nfbr 5077	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘2 ∥ ⦋𝑧 / 𝑘⦌𝐵
40	39	nfn 1858	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵
41		csbeq1a 3842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	breq2d 5042	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → (2 ∥ 𝐵 ↔ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
43	42	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (¬ 2 ∥ 𝐵 ↔ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
44	40, 43	rspc 3559	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ¬ 2 ∥ 𝐵 → ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
45	26, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → (∀𝑘 ∈ 𝐴 ¬ 2 ∥ 𝐵 → ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
46	35, 45	syl5com 31	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∖ 𝑦) → ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
47	46	a1d 25	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ⊆ 𝐴 → (𝑧 ∈ (𝐴 ∖ 𝑦) → ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)))
48	47	imp32 422	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)
49	33, 48	jca 515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
50	49	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
51		sumeven.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ Fin)
52		ssfi 8722	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
53	52	expcom 417	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝐴 → (𝐴 ∈ Fin → 𝑦 ∈ Fin))
54	53	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝐴 ∈ Fin → 𝑦 ∈ Fin))
55	51, 54	syl5com 31	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑦 ∈ Fin))
56	55	imp 410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
57		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
58		ssel 3908	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝐴 → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
59	58	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
60	59	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
61	60	imp 410	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
62	57, 61, 29	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℤ)
63	56, 62	fsumzcl 15084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ)
64	63	anim1i 617	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
65		opeo 15706	. . . . . . . . 9 ⊢ (((⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵) ∧ (Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵)) → ¬ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵))
66	50, 64, 65	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → ¬ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵))
67	63	zcnd 12076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
68	33	zcnd 12076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
69		addcom 10815	. . . . . . . . . . . 12 ⊢ ((Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵))
70	69	breq2d 5042	. . . . . . . . . . 11 ⊢ ((Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → (2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵)))
71	70	notbid 321	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ ¬ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵)))
72	67, 68, 71	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ ¬ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵)))
73	72	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ ¬ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵)))
74	66, 73	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
75	74	ex 416	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 → ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
76	63	anim1i 617	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
77	49	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
78		opoe 15704	. . . . . . . . 9 ⊢ (((Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) ∧ (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)) → 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
79	76, 77, 78	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
80	79	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵 → 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
81	80	con1d 147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) → 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
82	75, 81	impbid 215	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
83		bitr3 356	. . . . 5 ⊢ ((2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) → ((2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1))))
84	82, 83	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1))))
85		bicom 225	. . . 4 ⊢ ((¬ 2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) ↔ (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
86		bicom 225	. . . 4 ⊢ ((¬ 2 ∥ ((♯‘𝑦) + 1) ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) ↔ (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
87	84, 85, 86	3imtr4g 299	. . 3 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((¬ 2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (¬ 2 ∥ ((♯‘𝑦) + 1) ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))))
88		notnotb 318	. . . . 5 ⊢ (2 ∥ (♯‘𝑦) ↔ ¬ ¬ 2 ∥ (♯‘𝑦))
89		hashcl 13713	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
90	56, 89	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝑦) ∈ ℕ₀)
91	90	nn0zd 12073	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝑦) ∈ ℤ)
92		oddp1even 15685	. . . . . . 7 ⊢ ((♯‘𝑦) ∈ ℤ → (¬ 2 ∥ (♯‘𝑦) ↔ 2 ∥ ((♯‘𝑦) + 1)))
93	91, 92	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ 2 ∥ (♯‘𝑦) ↔ 2 ∥ ((♯‘𝑦) + 1)))
94	93	notbid 321	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ ¬ 2 ∥ (♯‘𝑦) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
95	88, 94	syl5bb 286	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ (♯‘𝑦) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
96	95	bibi1d 347	. . 3 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((2 ∥ (♯‘𝑦) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) ↔ (¬ 2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵)))
97		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
98		eldifn 4055	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → ¬ 𝑧 ∈ 𝑦)
99	98	adantl 485	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
100	99	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
101	56, 100	jca 515	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦))
102		hashunsng 13749	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
103	97, 101, 102	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
104	103	breq2d 5042	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ ((♯‘𝑦) + 1)))
105		vex 3444	. . . . . . . 8 ⊢ 𝑧 ∈ V
106	105	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ V)
107		df-nel 3092	. . . . . . . 8 ⊢ (𝑧 ∉ 𝑦 ↔ ¬ 𝑧 ∈ 𝑦)
108	100, 107	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∉ 𝑦)
109		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
110		elun 4076	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}))
111	59	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑦 → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
112		elsni 4542	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ {𝑧} → 𝑘 = 𝑧)
113		eleq1w 2872	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
114	27, 113	syl5ibr 249	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
115	112, 114	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {𝑧} → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
116	111, 115	jaoi 854	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
117	110, 116	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
118	117	com12 32	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝑘 ∈ (𝑦 ∪ {𝑧}) → 𝑘 ∈ 𝐴))
119	118	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ (𝑦 ∪ {𝑧}) → 𝑘 ∈ 𝐴))
120	119	imp 410	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴)
121	109, 120, 29	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℤ)
122	121	ralrimiva 3149	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ ℤ)
123		fsumsplitsnun 15102	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ (𝑧 ∈ V ∧ 𝑧 ∉ 𝑦) ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
124	56, 106, 108, 122, 123	syl121anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
125	124	breq2d 5042	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
126	104, 125	bibi12d 349	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ↔ (2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))))
127		notbi 322	. . . 4 ⊢ ((2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) ↔ (¬ 2 ∥ ((♯‘𝑦) + 1) ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
128	126, 127	syl6bb 290	. . 3 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ↔ (¬ 2 ∥ ((♯‘𝑦) + 1) ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))))
129	87, 96, 128	3imtr4d 297	. 2 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((2 ∥ (♯‘𝑦) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
130	9, 14, 19, 24, 25, 129, 51	findcard2d 8744	1 ⊢ (𝜑 → (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 ∀wral 3106 Vcvv 3441 ⦋csb 3828 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ♯chash 13686 Σcsu 15034 ∥ cdvds 15599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-dvds 15600

This theorem is referenced by: evensumodd 15730 oddsumodd 15731 vtxdgoddnumeven 27343

W3C validator