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Theorem sumodd 16425

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 6906	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
2		hash0 14406	. . . . 5 ⊢ (♯‘∅) = 0
3	1, 2	eqtrdi 2793	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
4	3	breq2d 5155	. . 3 ⊢ (𝑥 = ∅ → (2 ∥ (♯‘𝑥) ↔ 2 ∥ 0))
5		sumeq1 15725	. . . . 5 ⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
6		sum0 15757	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
7	5, 6	eqtrdi 2793	. . . 4 ⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 𝐵 = 0)
8	7	breq2d 5155	. . 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 6906	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
11	10	breq2d 5155	. . 3 ⊢ (𝑥 = 𝑦 → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘𝑦)))
12		sumeq1 15725	. . . 4 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
13	12	breq2d 5155	. . 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 6906	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	breq2d 5155	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘(𝑦 ∪ {𝑧}))))
17		sumeq1 15725	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
18	17	breq2d 5155	. . 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 6906	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
21	20	breq2d 5155	. . 3 ⊢ (𝑥 = 𝐴 → (2 ∥ (♯‘𝑥) ↔ 2 ∥ (♯‘𝐴)))
22		sumeq1 15725	. . . 4 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
23	22	breq2d 5155	. . 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 262	. 2 ⊢ (𝜑 → (2 ∥ 0 ↔ 2 ∥ 0))
26		eldifi 4131	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → 𝑧 ∈ 𝐴)
27	26	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑧 ∈ 𝐴)
28	27	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
29		sumeven.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)
30	29	adantlr 715	. . . . . . . . . . . . 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 4438	. . . . . . . . . . . 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 2905	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘2
37		nfcv 2905	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ∥
38		nfcsb1v 3923	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
39	36, 37, 38	nfbr 5190	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘2 ∥ ⦋𝑧 / 𝑘⦌𝐵
40	39	nfn 1857	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵
41		csbeq1a 3913	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	breq2d 5155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → (2 ∥ 𝐵 ↔ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
43	42	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (¬ 2 ∥ 𝐵 ↔ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
44	40, 43	rspc 3610	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)
49	33, 48	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
50	49	adantr 480	. . . . . . . . 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 9213	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
53	52	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝐴 → (𝐴 ∈ Fin → 𝑦 ∈ Fin))
54	53	adantr 480	. . . . . . . . . . . . 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 406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
57		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
58		ssel 3977	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝐴 → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
59	58	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
60	59	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
61	60	imp 406	. . . . . . . . . . . 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 15771	. . . . . . . . . 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 16402	. . . . . . . . 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 12723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
68	33	zcnd 12723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
69		addcom 11447	. . . . . . . . . . . 12 ⊢ ((Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (⦋𝑧 / 𝑘⦌𝐵 + Σ𝑘 ∈ 𝑦 𝐵))
70	69	breq2d 5155	. . . . . . . . . . 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 318	. . . . . . . . . 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 480	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → ¬ 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
75	74	ex 412	. . . . . 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 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ¬ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ ∧ ¬ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))
78		opoe 16400	. . . . . . . . 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 412	. . . . . . 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 212	. . . . 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 222	. . . 4 ⊢ ((¬ 2 ∥ ((♯‘𝑦) + 1) ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) ↔ (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
86		bicom 222	. . . 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 296	. . 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 315	. . . . 5 ⊢ (2 ∥ (♯‘𝑦) ↔ ¬ ¬ 2 ∥ (♯‘𝑦))
89		hashcl 14395	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
90	56, 89	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝑦) ∈ ℕ₀)
91	90	nn0zd 12639	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝑦) ∈ ℤ)
92		oddp1even 16381	. . . . . . 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 318	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (¬ ¬ 2 ∥ (♯‘𝑦) ↔ ¬ 2 ∥ ((♯‘𝑦) + 1)))
95	88, 94	bitrid 283	. . . 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 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
98		eldifn 4132	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → ¬ 𝑧 ∈ 𝑦)
99	98	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
100	99	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
101	56, 100	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦))
102		hashunsng 14431	. . . . . . 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 5155	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ (♯‘(𝑦 ∪ {𝑧})) ↔ 2 ∥ ((♯‘𝑦) + 1)))
105		vex 3484	. . . . . . . 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 234	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∉ 𝑦)
109		simpll 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
110		elun 4153	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}))
111	59	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑦 → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
112		elsni 4643	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ {𝑧} → 𝑘 = 𝑧)
113		eleq1w 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
114	27, 113	imbitrrid 246	. . . . . . . . . . . . . . 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 858	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴))
117	110, 116	sylbi 217	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ (𝑦 ∪ {𝑧}) → 𝑘 ∈ 𝐴))
120	119	imp 406	. . . . . . . . 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 15791	. . . . . . 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 1377	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
125	124	breq2d 5155	. . . . 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 319	. . . 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 287	. . 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 294	. 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 9206	1 ⊢ (𝜑 → (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 ∀wral 3061 Vcvv 3480 ⦋csb 3899 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ♯chash 14369 Σcsu 15722 ∥ cdvds 16290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-dvds 16291

This theorem is referenced by: evensumodd 16426 oddsumodd 16427 vtxdgoddnumeven 29571

W3C validator