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Theorem sumodd 16327

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 6888	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
2		hash0 14323	. . . . 5 ⊢ (♯‘∅) = 0
3	1, 2	eqtrdi 2788	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
4	3	breq2d 5159	. . 3 ⊢ (𝑥 = ∅ → (2 ∥ (♯‘𝑥) ↔ 2 ∥ 0))
5		sumeq1 15631	. . . . 5 ⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
6		sum0 15663	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
7	5, 6	eqtrdi 2788	. . . 4 ⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 𝐵 = 0)
8	7	breq2d 5159	. . 3 ⊢ (𝑥 = ∅ → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ 0))
9	4, 8	bibi12d 345	. 2 ⊢ (𝑥 = ∅ → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ 0 ↔ 2 ∥ 0)))
10		fveq2 6888	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
11	10	breq2d 5159	. . 3 ⊢ (𝑥 = 𝑦 → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘𝑦)))
12		sumeq1 15631	. . . 4 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
13	12	breq2d 5159	. . 3 ⊢ (𝑥 = 𝑦 → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
14	11, 13	bibi12d 345	. 2 ⊢ (𝑥 = 𝑦 → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ (♯‘𝑦) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵)))
15		fveq2 6888	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	breq2d 5159	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘(𝑦 ∪ {𝑧}))))
17		sumeq1 15631	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
18	17	breq2d 5159	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
19	16, 18	bibi12d 345	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
20		fveq2 6888	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
21	20	breq2d 5159	. . 3 ⊢ (𝑥 = 𝐴 → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘𝐴)))
22		sumeq1 15631	. . . 4 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
23	22	breq2d 5159	. . 3 ⊢ (𝑥 = 𝐴 → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵))
24	21, 23	bibi12d 345	. 2 ⊢ (𝑥 = 𝐴 → ((2 ∥ (♯‘𝑥) ↔ 2 ∥ Σ𝑘 ∈ 𝑥 𝐵) ↔ (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵)))
25		biidd 261	. 2 ⊢ (𝜑 → (2 ∥ 0 ↔ 2 ∥ 0))
26		eldifi 4125	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → 𝑧 ∈ 𝐴)
27	26	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑧 ∈ 𝐴)
28	27	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
29		sumeven.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)
30	29	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)
31	30	ralrimiva 3146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ)
32		rspcsbela 4434	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
33	28, 31, 32	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
34		sumodd.o	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵)
35	34	ralrimiva 3146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ¬ 2 ∥ 𝐵)
36		nfcv 2903	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘2
37		nfcv 2903	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ∥
38		nfcsb1v 3917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
39	36, 37, 38	nfbr 5194	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘2 ∥ ⦋𝑧 / 𝑘⦌𝐵
40	39	nfn 1860	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵
41		csbeq1a 3906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	breq2d 5159	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → (2 ∥ 𝐵 ↔ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
43	42	notbid 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (¬ 2 ∥ 𝐵 ↔ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
44	40, 43	rspc 3600	. . . . . . . . . . . . . . 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 419	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)
49	33, 48	jca 512	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
50	49	adantr 481	. . . . . . . . 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 9169	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
53	52	expcom 414	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝐴 → (𝐴 ∈ Fin → 𝑦 ∈ Fin))
54	53	adantr 481	. . . . . . . . . . . . 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 407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
57		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
58		ssel 3974	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝐴 → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
59	58	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
60	59	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
61	60	imp 407	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
62	57, 61, 29	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℤ)
63	56, 62	fsumzcl 15677	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ)
64	63	anim1i 615	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
65		opeo 16304	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → ¬ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵))
67	63	zcnd 12663	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
68	33	zcnd 12663	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
69		addcom 11396	. . . . . . . . . . . 12 ⊢ ((Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵))
70	69	breq2d 5159	. . . . . . . . . . 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 317	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ↔ ¬ 2 ∥ (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵)))
73	72	adantr 481	. . . . . . . 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 256	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
75	74	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 → ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
76	63	anim1i 615	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
77	49	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
78		opoe 16302	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
80	79	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵 → 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
81	80	con1d 145	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) → 2 ∥ Σ𝑘 ∈ 𝑦 𝐵))
82	75, 81	impbid 211	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
83		bitr3 352	. . . . 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 221	. . . 4 ⊢ ((¬ 2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) ↔ (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
86		bicom 221	. . . 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 295	. . 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 314	. . . . 5 ⊢ (2 ∥ (♯‘𝑦) ↔ ¬ ¬ 2 ∥ (♯‘𝑦))
89		hashcl 14312	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
90	56, 89	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝑦) ∈ ℕ₀)
91	90	nn0zd 12580	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝑦) ∈ ℤ)
92		oddp1even 16283	. . . . . . 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 317	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ ¬ 2 ∥ (♯‘𝑦) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
95	88, 94	bitrid 282	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ (♯‘𝑦) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
96	95	bibi1d 343	. . 3 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((2 ∥ (♯‘𝑦) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) ↔ (¬ 2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵)))
97		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
98		eldifn 4126	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → ¬ 𝑧 ∈ 𝑦)
99	98	adantl 482	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
100	99	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
101	56, 100	jca 512	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦))
102		hashunsng 14348	. . . . . . 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 5159	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ ((♯‘𝑦) + 1)))
105		vex 3478	. . . . . . . 8 ⊢ 𝑧 ∈ V
106	105	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ V)
107		df-nel 3047	. . . . . . . 8 ⊢ (𝑧 ∉ 𝑦 ↔ ¬ 𝑧 ∈ 𝑦)
108	100, 107	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∉ 𝑦)
109		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
110		elun 4147	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}))
111	59	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑦 → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
112		elsni 4644	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ {𝑧} → 𝑘 = 𝑧)
113		eleq1w 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
114	27, 113	imbitrrid 245	. . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
117	110, 116	sylbi 216	. . . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ (𝑦 ∪ {𝑧}) → 𝑘 ∈ 𝐴))
120	119	imp 407	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴)
121	109, 120, 29	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℤ)
122	121	ralrimiva 3146	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ ℤ)
123		fsumsplitsnun 15697	. . . . . . 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 1375	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
125	124	breq2d 5159	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
126	104, 125	bibi12d 345	. . . 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 318	. . . 4 ⊢ ((2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) ↔ (¬ 2 ∥ ((♯‘𝑦) + 1) ↔ ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
128	126, 127	bitrdi 286	. . 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 293	. 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 9162	1 ⊢ (𝜑 → (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∉ wnel 3046 ∀wral 3061 Vcvv 3474 ⦋csb 3892 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 ∅c0 4321 {csn 4627 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ♯chash 14286 Σcsu 15628 ∥ cdvds 16193

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-dvds 16194

This theorem is referenced by: evensumodd 16328 oddsumodd 16329 vtxdgoddnumeven 28799

W3C validator