Theorem axlowdimlem15 28971

Description: Lemma for axlowdim 28976. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)

Hypothesis

Ref	Expression
axlowdimlem15.1	⊢ 𝐹 = (𝑖 ∈ (1...(𝑁 − 1)) ↦ if(𝑖 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))))

Assertion

Ref	Expression
axlowdimlem15	⊢ (𝑁 ∈ (ℤ≥‘3) → 𝐹:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁))

Distinct variable group:   𝑖,𝑁

Allowed substitution hint:   𝐹(𝑖)

Proof of Theorem axlowdimlem15

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . . 6 ⊢ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
2	1	axlowdimlem7 28963	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
3	2	adantr 480	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
4		eluzge3nn 12932	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
5		eqid 2737	. . . . . 6 ⊢ ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))
6	5	axlowdimlem10 28966	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
7	4, 6	sylan 580	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
8	3, 7	ifcld 4572	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))) ∈ (𝔼‘𝑁))
9		axlowdimlem15.1	. . 3 ⊢ 𝐹 = (𝑖 ∈ (1...(𝑁 − 1)) ↦ if(𝑖 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))))
10	8, 9	fmptd 7134	. 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝐹:(1...(𝑁 − 1))⟶(𝔼‘𝑁))
11		eqeq1 2741	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑖 = 1 ↔ 𝑗 = 1))
12		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
13	12	opeq1d 4879	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑗 + 1), 1⟩)
14	13	sneqd 4638	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑗 + 1), 1⟩})
15	12	sneqd 4638	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → {(𝑖 + 1)} = {(𝑗 + 1)})
16	15	difeq2d 4126	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑗 + 1)}))
17	16	xpeq1d 5714	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
18	14, 17	uneq12d 4169	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
19	11, 18	ifbieq2d 4552	. . . . . . 7 ⊢ (𝑖 = 𝑗 → if(𝑖 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))) = if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))))
20		snex 5436	. . . . . . . . 9 ⊢ {⟨3, -1⟩} ∈ V
21		ovex 7464	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ V
22	21	difexi 5330	. . . . . . . . . 10 ⊢ ((1...𝑁) ∖ {3}) ∈ V
23		snex 5436	. . . . . . . . . 10 ⊢ {0} ∈ V
24	22, 23	xpex 7773	. . . . . . . . 9 ⊢ (((1...𝑁) ∖ {3}) × {0}) ∈ V
25	20, 24	unex 7764	. . . . . . . 8 ⊢ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ V
26		snex 5436	. . . . . . . . 9 ⊢ {⟨(𝑗 + 1), 1⟩} ∈ V
27	21	difexi 5330	. . . . . . . . . 10 ⊢ ((1...𝑁) ∖ {(𝑗 + 1)}) ∈ V
28	27, 23	xpex 7773	. . . . . . . . 9 ⊢ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}) ∈ V
29	26, 28	unex 7764	. . . . . . . 8 ⊢ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ∈ V
30	25, 29	ifex 4576	. . . . . . 7 ⊢ if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) ∈ V
31	19, 9, 30	fvmpt 7016	. . . . . 6 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝐹‘𝑗) = if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))))
32		eqeq1 2741	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝑖 = 1 ↔ 𝑘 = 1))
33		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
34	33	opeq1d 4879	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑘 + 1), 1⟩)
35	34	sneqd 4638	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑘 + 1), 1⟩})
36	33	sneqd 4638	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → {(𝑖 + 1)} = {(𝑘 + 1)})
37	36	difeq2d 4126	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑘 + 1)}))
38	37	xpeq1d 5714	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
39	35, 38	uneq12d 4169	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
40	32, 39	ifbieq2d 4552	. . . . . . 7 ⊢ (𝑖 = 𝑘 → if(𝑖 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))))
41		snex 5436	. . . . . . . . 9 ⊢ {⟨(𝑘 + 1), 1⟩} ∈ V
42	21	difexi 5330	. . . . . . . . . 10 ⊢ ((1...𝑁) ∖ {(𝑘 + 1)}) ∈ V
43	42, 23	xpex 7773	. . . . . . . . 9 ⊢ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}) ∈ V
44	41, 43	unex 7764	. . . . . . . 8 ⊢ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) ∈ V
45	25, 44	ifex 4576	. . . . . . 7 ⊢ if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ∈ V
46	40, 9, 45	fvmpt 7016	. . . . . 6 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → (𝐹‘𝑘) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))))
47	31, 46	eqeqan12d 2751	. . . . 5 ⊢ ((𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝐹‘𝑗) = (𝐹‘𝑘) ↔ if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))))
48	47	adantl 481	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → ((𝐹‘𝑗) = (𝐹‘𝑘) ↔ if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))))
49		eqtr3 2763	. . . . . 6 ⊢ ((𝑗 = 1 ∧ 𝑘 = 1) → 𝑗 = 𝑘)
50	49	2a1d 26	. . . . 5 ⊢ ((𝑗 = 1 ∧ 𝑘 = 1) → ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘)))
51		eqid 2737	. . . . . . . . . . 11 ⊢ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
52	1, 51	axlowdimlem13 28969	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
53	52	neneqd 2945	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
54	53	pm2.21d 121	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
55	54	adantrl 716	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
56	4, 55	sylan 580	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
57		iftrue 4531	. . . . . . . 8 ⊢ (𝑗 = 1 → if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
58		iffalse 4534	. . . . . . . 8 ⊢ (¬ 𝑘 = 1 → if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
59	57, 58	eqeqan12d 2751	. . . . . . 7 ⊢ ((𝑗 = 1 ∧ ¬ 𝑘 = 1) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ↔ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))))
60	59	imbi1d 341	. . . . . 6 ⊢ ((𝑗 = 1 ∧ ¬ 𝑘 = 1) → ((if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘) ↔ (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘)))
61	56, 60	imbitrrid 246	. . . . 5 ⊢ ((𝑗 = 1 ∧ ¬ 𝑘 = 1) → ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘)))
62		eqid 2737	. . . . . . . . . . . 12 ⊢ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
63	1, 62	axlowdimlem13 28969	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
64	63	necomd 2996	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ≠ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
65	64	neneqd 2945	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ¬ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
66	65	pm2.21d 121	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘))
67	4, 66	sylan 580	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘))
68	67	adantrr 717	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘))
69		iffalse 4534	. . . . . . . 8 ⊢ (¬ 𝑗 = 1 → if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
70		iftrue 4531	. . . . . . . 8 ⊢ (𝑘 = 1 → if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
71	69, 70	eqeqan12d 2751	. . . . . . 7 ⊢ ((¬ 𝑗 = 1 ∧ 𝑘 = 1) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ↔ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))))
72	71	imbi1d 341	. . . . . 6 ⊢ ((¬ 𝑗 = 1 ∧ 𝑘 = 1) → ((if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘) ↔ (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘)))
73	68, 72	imbitrrid 246	. . . . 5 ⊢ ((¬ 𝑗 = 1 ∧ 𝑘 = 1) → ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘)))
74	62, 51	axlowdimlem14 28970	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
75	74	3expb 1121	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
76	4, 75	sylan 580	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
77	69, 58	eqeqan12d 2751	. . . . . . 7 ⊢ ((¬ 𝑗 = 1 ∧ ¬ 𝑘 = 1) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ↔ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))))
78	77	imbi1d 341	. . . . . 6 ⊢ ((¬ 𝑗 = 1 ∧ ¬ 𝑘 = 1) → ((if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘) ↔ (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘)))
79	76, 78	imbitrrid 246	. . . . 5 ⊢ ((¬ 𝑗 = 1 ∧ ¬ 𝑘 = 1) → ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘)))
80	50, 61, 73, 79	4cases 1041	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → (if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) → 𝑗 = 𝑘))
81	48, 80	sylbid 240	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → ((𝐹‘𝑗) = (𝐹‘𝑘) → 𝑗 = 𝑘))
82	81	ralrimivva 3202	. 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → ∀𝑗 ∈ (1...(𝑁 − 1))∀𝑘 ∈ (1...(𝑁 − 1))((𝐹‘𝑗) = (𝐹‘𝑘) → 𝑗 = 𝑘))
83		dff13 7275	. 2 ⊢ (𝐹:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁) ↔ (𝐹:(1...(𝑁 − 1))⟶(𝔼‘𝑁) ∧ ∀𝑗 ∈ (1...(𝑁 − 1))∀𝑘 ∈ (1...(𝑁 − 1))((𝐹‘𝑗) = (𝐹‘𝑘) → 𝑗 = 𝑘)))
84	10, 82, 83	sylanbrc 583	1 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝐹:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ∖ cdif 3948   ∪ cun 3949  ifcif 4525  {csn 4626  ⟨cop 4632   ↦ cmpt 5225   × cxp 5683  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158   − cmin 11492  -cneg 11493  ℕcn 12266  3c3 12322  ℤ_≥cuz 12878  ...cfz 13547  𝔼cee 28903

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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  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

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-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-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-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-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-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-ee 28906

This theorem is referenced by:  axlowdim  28976