Theorem ballotlemsf1o 33202

Description: The defined 𝑆 is a bijection, and an involution. (Contributed by Thierry Arnoux, 14-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotth.o	⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p	⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))

Assertion

Ref	Expression
ballotlemsf1o	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝐶,𝑖,𝑘   𝑖,𝐸,𝑘   𝐶,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐

Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑆(𝑥,𝑖,𝑘,𝑐)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemsf1o

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . 5 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . 5 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . . . 5 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . . 5 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . . 5 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . . 5 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . . 5 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . . 5 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsval 33197	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))
11	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsv 33198	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑖) = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖))
12	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsdom 33200	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
13	11, 12	eqeltrrd 2833	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖) ∈ (1...(𝑀 + 𝑁)))
14	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsv 33198	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑗) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
15	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsdom 33200	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
16	14, 15	eqeltrrd 2833	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) ∈ (1...(𝑀 + 𝑁)))
17		oveq2 7370	. . . . . 6 ⊢ (𝑖 = (((𝐼‘𝐶) + 1) − 𝑗) → (((𝐼‘𝐶) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑗)))
18		id 22	. . . . . 6 ⊢ (𝑖 = 𝑗 → 𝑖 = 𝑗)
19		breq1 5113	. . . . . 6 ⊢ (𝑖 = (((𝐼‘𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼‘𝐶) ↔ (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶)))
20		breq1 5113	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ≤ (𝐼‘𝐶) ↔ 𝑗 ≤ (𝐼‘𝐶)))
21	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 33190	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
22	21	simpld 495	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
23		elfzelz 13451	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
24	23	peano2zd 12619	. . . . . . . . . . 11 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → ((𝐼‘𝐶) + 1) ∈ ℤ)
25	22, 24	syl 17	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) + 1) ∈ ℤ)
26	25	zcnd 12617	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) + 1) ∈ ℂ)
27	26	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼‘𝐶) + 1) ∈ ℂ)
28		elfzelz 13451	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
29	28	zcnd 12617	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
30	29	ad2antll 727	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
31	27, 30	nncand 11526	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑗)) = 𝑗)
32	31	eqcomd 2737	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑗)))
33	22, 23	syl 17	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
34	33	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝐼‘𝐶) ∈ ℤ)
35		elfznn 13480	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
36	35	ad2antll 727	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
37	34, 36	ltesubnnd 31788	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶))
38	37	adantr 481	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 ≤ (𝐼‘𝐶)) → (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶))
39		vex 3450	. . . . . . 7 ⊢ 𝑗 ∈ V
40	39	a1i 11	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ V)
41		ovexd 7397	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − 𝑗) ∈ V)
42	17, 18, 19, 20, 32, 38, 40, 41	ifeqeqx 31528	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)) → 𝑗 = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖))
43		oveq2 7370	. . . . . 6 ⊢ (𝑗 = (((𝐼‘𝐶) + 1) − 𝑖) → (((𝐼‘𝐶) + 1) − 𝑗) = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)))
44		id 22	. . . . . 6 ⊢ (𝑗 = 𝑖 → 𝑗 = 𝑖)
45		breq1 5113	. . . . . 6 ⊢ (𝑗 = (((𝐼‘𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼‘𝐶) ↔ (((𝐼‘𝐶) + 1) − 𝑖) ≤ (𝐼‘𝐶)))
46		breq1 5113	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ≤ (𝐼‘𝐶) ↔ 𝑖 ≤ (𝐼‘𝐶)))
47		elfzelz 13451	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
48	47	zcnd 12617	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
49	48	ad2antrl 726	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
50	27, 49	nncand 11526	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)) = 𝑖)
51	50	eqcomd 2737	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)))
52	34	adantr 481	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℤ)
53		simplrl 775	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → 𝑖 ∈ (1...(𝑀 + 𝑁)))
54		elfznn 13480	. . . . . . . 8 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℕ)
55	53, 54	syl 17	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → 𝑖 ∈ ℕ)
56	52, 55	ltesubnnd 31788	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → (((𝐼‘𝐶) + 1) − 𝑖) ≤ (𝐼‘𝐶))
57		vex 3450	. . . . . . 7 ⊢ 𝑖 ∈ V
58	57	a1i 11	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ V)
59		ovexd 7397	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − 𝑖) ∈ V)
60	43, 44, 45, 46, 51, 56, 58, 59	ifeqeqx 31528	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) → 𝑖 = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
61	42, 60	impbida 799	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝑖 = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) ↔ 𝑗 = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))
62	10, 13, 16, 61	f1o3d 31608	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))))
63	62	simpld 495	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
64		oveq2 7370	. . . . . 6 ⊢ (𝑖 = 𝑗 → (((𝐼‘𝐶) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑗))
65	20, 64, 18	ifbieq12d 4519	. . . . 5 ⊢ (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
66	65	cbvmptv 5223	. . . 4 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
67	66	a1i 11	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)))
68	62	simprd 496	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ◡(𝑆‘𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)))
69	67, 10, 68	3eqtr4rd 2782	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ◡(𝑆‘𝐶) = (𝑆‘𝐶))
70	63, 69	jca 512	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {crab 3405  Vcvv 3446   ∖ cdif 3910   ∩ cin 3912  ifcif 4491  𝒫 cpw 4565   class class class wbr 5110   ↦ cmpt 5193  ^◡ccnv 5637  –1-1-onto→wf1o 6500  ‘cfv 6501  (class class class)co 7362  infcinf 9386  ℂcc 11058  ℝcr 11059  0cc0 11060  1c1 11061   + caddc 11063   < clt 11198   ≤ cle 11199   − cmin 11394   / cdiv 11821  ℕcn 12162  ℤcz 12508  ...cfz 13434  ♯chash 14240

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9387  df-inf 9388  df-dju 9846  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-fz 13435  df-hash 14241

This theorem is referenced by:  ballotlemsima  33204  ballotlemscr  33207  ballotlemrv  33208  ballotlemro  33211  ballotlemfrc  33215  ballotlemrinv0  33221