Theorem axlowdimlem15 27612

Description: Lemma for axlowdim 27617. 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 27604	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
3	2	adantr 482	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
4		eluzge3nn 12735	. . . . 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 27607	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
7	4, 6	sylan 581	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
8	3, 7	ifcld 4523	. . 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 7048	. 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝐹:(1...(𝑁 − 1))⟶(𝔼‘𝑁))
11		eqeq1 2741	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑖 = 1 ↔ 𝑗 = 1))
12		oveq1 7348	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
13	12	opeq1d 4827	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑗 + 1), 1⟩)
14	13	sneqd 4589	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑗 + 1), 1⟩})
15	12	sneqd 4589	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → {(𝑖 + 1)} = {(𝑗 + 1)})
16	15	difeq2d 4073	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑗 + 1)}))
17	16	xpeq1d 5653	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
18	14, 17	uneq12d 4115	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
19	11, 18	ifbieq2d 4503	. . . . . . 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 5380	. . . . . . . . 9 ⊢ {⟨3, -1⟩} ∈ V
21		ovex 7374	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ V
22	21	difexi 5276	. . . . . . . . . 10 ⊢ ((1...𝑁) ∖ {3}) ∈ V
23		snex 5380	. . . . . . . . . 10 ⊢ {0} ∈ V
24	22, 23	xpex 7669	. . . . . . . . 9 ⊢ (((1...𝑁) ∖ {3}) × {0}) ∈ V
25	20, 24	unex 7662	. . . . . . . 8 ⊢ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ V
26		snex 5380	. . . . . . . . 9 ⊢ {⟨(𝑗 + 1), 1⟩} ∈ V
27	21	difexi 5276	. . . . . . . . . 10 ⊢ ((1...𝑁) ∖ {(𝑗 + 1)}) ∈ V
28	27, 23	xpex 7669	. . . . . . . . 9 ⊢ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}) ∈ V
29	26, 28	unex 7662	. . . . . . . 8 ⊢ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ∈ V
30	25, 29	ifex 4527	. . . . . . 7 ⊢ if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) ∈ V
31	19, 9, 30	fvmpt 6935	. . . . . 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 7348	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
34	33	opeq1d 4827	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑘 + 1), 1⟩)
35	34	sneqd 4589	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑘 + 1), 1⟩})
36	33	sneqd 4589	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → {(𝑖 + 1)} = {(𝑘 + 1)})
37	36	difeq2d 4073	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑘 + 1)}))
38	37	xpeq1d 5653	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
39	35, 38	uneq12d 4115	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
40	32, 39	ifbieq2d 4503	. . . . . . 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 5380	. . . . . . . . 9 ⊢ {⟨(𝑘 + 1), 1⟩} ∈ V
42	21	difexi 5276	. . . . . . . . . 10 ⊢ ((1...𝑁) ∖ {(𝑘 + 1)}) ∈ V
43	42, 23	xpex 7669	. . . . . . . . 9 ⊢ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}) ∈ V
44	41, 43	unex 7662	. . . . . . . 8 ⊢ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) ∈ V
45	25, 44	ifex 4527	. . . . . . 7 ⊢ if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ∈ V
46	40, 9, 45	fvmpt 6935	. . . . . 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 483	. . . 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 27610	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
53	52	neneqd 2946	. . . . . . . . 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 714	. . . . . . 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 581	. . . . . 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 4483	. . . . . . . 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 4486	. . . . . . . 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 342	. . . . . 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	syl5ibr 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 27610	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
64	63	necomd 2997	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ≠ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
65	64	neneqd 2946	. . . . . . . . 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 581	. . . . . . 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 715	. . . . . 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 4486	. . . . . . . 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 4483	. . . . . . . 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 342	. . . . . 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	syl5ibr 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 27611	. . . . . . . 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 1120	. . . . . . 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 581	. . . . . 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 342	. . . . . 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	syl5ibr 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 1039	. . . 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 239	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → ((𝐹‘𝑗) = (𝐹‘𝑘) → 𝑗 = 𝑘))
82	81	ralrimivva 3194	. 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → ∀𝑗 ∈ (1...(𝑁 − 1))∀𝑘 ∈ (1...(𝑁 − 1))((𝐹‘𝑗) = (𝐹‘𝑘) → 𝑗 = 𝑘))
83		dff13 7188	. 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 584	1 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝐹:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ∀wral 3062   ∖ cdif 3898   ∪ cun 3899  ifcif 4477  {csn 4577  ⟨cop 4583   ↦ cmpt 5179   × cxp 5622  ⟶wf 6479  –1-1→wf1 6480  ‘cfv 6483  (class class class)co 7341  0cc0 10976  1c1 10977   + caddc 10979   − cmin 11310  -cneg 11311  ℕcn 12078  3c3 12134  ℤ_≥cuz 12687  ...cfz 13344  𝔼cee 27544

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 2708  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053

This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-er 8573  df-map 8692  df-en 8809  df-dom 8810  df-sdom 8811  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-nn 12079  df-2 12141  df-3 12142  df-n0 12339  df-z 12425  df-uz 12688  df-fz 13345  df-ee 27547

This theorem is referenced by:  axlowdim  27617