Theorem ballotlemsf1o 34478

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 34473	. . . 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 34474	. . . . 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 34476	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
13	11, 12	eqeltrrd 2845	. . . 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 34474	. . . . 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 34476	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
16	14, 15	eqeltrrd 2845	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) ∈ (1...(𝑀 + 𝑁)))
17		oveq2 7456	. . . . . 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 5169	. . . . . 6 ⊢ (𝑖 = (((𝐼‘𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼‘𝐶) ↔ (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶)))
20		breq1 5169	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ≤ (𝐼‘𝐶) ↔ 𝑗 ≤ (𝐼‘𝐶)))
21	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 34466	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
22	21	simpld 494	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
23		elfzelz 13584	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
24	23	peano2zd 12750	. . . . . . . . . . 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 12748	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) + 1) ∈ ℂ)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼‘𝐶) + 1) ∈ ℂ)
28		elfzelz 13584	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
29	28	zcnd 12748	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
30	29	ad2antll 728	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
31	27, 30	nncand 11652	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑗)) = 𝑗)
32	31	eqcomd 2746	. . . . . 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 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝐼‘𝐶) ∈ ℤ)
35		elfznn 13613	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
36	35	ad2antll 728	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
37	34, 36	ltesubnnd 32826	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶))
38	37	adantr 480	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 ≤ (𝐼‘𝐶)) → (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶))
39		vex 3492	. . . . . . 7 ⊢ 𝑗 ∈ V
40	39	a1i 11	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ V)
41		ovexd 7483	. . . . . 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 32565	. . . . 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 7456	. . . . . 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 5169	. . . . . 6 ⊢ (𝑗 = (((𝐼‘𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼‘𝐶) ↔ (((𝐼‘𝐶) + 1) − 𝑖) ≤ (𝐼‘𝐶)))
46		breq1 5169	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ≤ (𝐼‘𝐶) ↔ 𝑖 ≤ (𝐼‘𝐶)))
47		elfzelz 13584	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
48	47	zcnd 12748	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
49	48	ad2antrl 727	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
50	27, 49	nncand 11652	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)) = 𝑖)
51	50	eqcomd 2746	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)))
52	34	adantr 480	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℤ)
53		simplrl 776	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → 𝑖 ∈ (1...(𝑀 + 𝑁)))
54		elfznn 13613	. . . . . . . 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 32826	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → (((𝐼‘𝐶) + 1) − 𝑖) ≤ (𝐼‘𝐶))
57		vex 3492	. . . . . . 7 ⊢ 𝑖 ∈ V
58	57	a1i 11	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ V)
59		ovexd 7483	. . . . . 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 32565	. . . . 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 800	. . . 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 32646	. . 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 494	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
64		oveq2 7456	. . . . . 6 ⊢ (𝑖 = 𝑗 → (((𝐼‘𝐶) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑗))
65	20, 64, 18	ifbieq12d 4576	. . . . 5 ⊢ (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
66	65	cbvmptv 5279	. . . 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 495	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ◡(𝑆‘𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)))
69	67, 10, 68	3eqtr4rd 2791	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ◡(𝑆‘𝐶) = (𝑆‘𝐶))
70	63, 69	jca 511	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975  ifcif 4548  𝒫 cpw 4622   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  infcinf 9510  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325   − cmin 11520   / cdiv 11947  ℕcn 12293  ℤcz 12639  ...cfz 13567  ♯chash 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-hash 14380

This theorem is referenced by:  ballotlemsima  34480  ballotlemscr  34483  ballotlemrv  34484  ballotlemro  34487  ballotlemfrc  34491  ballotlemrinv0  34497