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Theorem axlowdimlem15 25636
 Description: Lemma for axlowdim 25641. 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.
StepHypRef Expression
1 eqid 2610 . . . . . 6 ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
21axlowdimlem7 25628 . . . . 5 (𝑁 ∈ (ℤ‘3) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
32adantr 480 . . . 4 ((𝑁 ∈ (ℤ‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
4 eluzge3nn 11606 . . . . 5 (𝑁 ∈ (ℤ‘3) → 𝑁 ∈ ℕ)
5 eqid 2610 . . . . . 6 ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))
65axlowdimlem10 25631 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
74, 6sylan 487 . . . 4 ((𝑁 ∈ (ℤ‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
83, 7ifcld 4081 . . 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}))))
108, 9fmptd 6292 . 2 (𝑁 ∈ (ℤ‘3) → 𝐹:(1...(𝑁 − 1))⟶(𝔼‘𝑁))
11 eqeq1 2614 . . . . . . . 8 (𝑖 = 𝑗 → (𝑖 = 1 ↔ 𝑗 = 1))
12 oveq1 6556 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
1312opeq1d 4346 . . . . . . . . . 10 (𝑖 = 𝑗 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑗 + 1), 1⟩)
1413sneqd 4137 . . . . . . . . 9 (𝑖 = 𝑗 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑗 + 1), 1⟩})
1512sneqd 4137 . . . . . . . . . . 11 (𝑖 = 𝑗 → {(𝑖 + 1)} = {(𝑗 + 1)})
1615difeq2d 3690 . . . . . . . . . 10 (𝑖 = 𝑗 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑗 + 1)}))
1716xpeq1d 5062 . . . . . . . . 9 (𝑖 = 𝑗 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
1814, 17uneq12d 3730 . . . . . . . 8 (𝑖 = 𝑗 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
1911, 18ifbieq2d 4061 . . . . . . 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 4835 . . . . . . . . 9 {⟨3, -1⟩} ∈ V
21 ovex 6577 . . . . . . . . . . 11 (1...𝑁) ∈ V
22 difexg 4735 . . . . . . . . . . 11 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {3}) ∈ V)
2321, 22ax-mp 5 . . . . . . . . . 10 ((1...𝑁) ∖ {3}) ∈ V
24 snex 4835 . . . . . . . . . 10 {0} ∈ V
2523, 24xpex 6860 . . . . . . . . 9 (((1...𝑁) ∖ {3}) × {0}) ∈ V
2620, 25unex 6854 . . . . . . . 8 ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ V
27 snex 4835 . . . . . . . . 9 {⟨(𝑗 + 1), 1⟩} ∈ V
28 difexg 4735 . . . . . . . . . . 11 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {(𝑗 + 1)}) ∈ V)
2921, 28ax-mp 5 . . . . . . . . . 10 ((1...𝑁) ∖ {(𝑗 + 1)}) ∈ V
3029, 24xpex 6860 . . . . . . . . 9 (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}) ∈ V
3127, 30unex 6854 . . . . . . . 8 ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ∈ V
3226, 31ifex 4106 . . . . . . 7 if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) ∈ V
3319, 9, 32fvmpt 6191 . . . . . 6 (𝑗 ∈ (1...(𝑁 − 1)) → (𝐹𝑗) = if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))))
34 eqeq1 2614 . . . . . . . 8 (𝑖 = 𝑘 → (𝑖 = 1 ↔ 𝑘 = 1))
35 oveq1 6556 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
3635opeq1d 4346 . . . . . . . . . 10 (𝑖 = 𝑘 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑘 + 1), 1⟩)
3736sneqd 4137 . . . . . . . . 9 (𝑖 = 𝑘 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑘 + 1), 1⟩})
3835sneqd 4137 . . . . . . . . . . 11 (𝑖 = 𝑘 → {(𝑖 + 1)} = {(𝑘 + 1)})
3938difeq2d 3690 . . . . . . . . . 10 (𝑖 = 𝑘 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑘 + 1)}))
4039xpeq1d 5062 . . . . . . . . 9 (𝑖 = 𝑘 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
4137, 40uneq12d 3730 . . . . . . . 8 (𝑖 = 𝑘 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
4234, 41ifbieq2d 4061 . . . . . . 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}))))
43 snex 4835 . . . . . . . . 9 {⟨(𝑘 + 1), 1⟩} ∈ V
44 difexg 4735 . . . . . . . . . . 11 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {(𝑘 + 1)}) ∈ V)
4521, 44ax-mp 5 . . . . . . . . . 10 ((1...𝑁) ∖ {(𝑘 + 1)}) ∈ V
4645, 24xpex 6860 . . . . . . . . 9 (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}) ∈ V
4743, 46unex 6854 . . . . . . . 8 ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) ∈ V
4826, 47ifex 4106 . . . . . . 7 if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ∈ V
4942, 9, 48fvmpt 6191 . . . . . 6 (𝑘 ∈ (1...(𝑁 − 1)) → (𝐹𝑘) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))))
5033, 49eqeqan12d 2626 . . . . 5 ((𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝐹𝑗) = (𝐹𝑘) ↔ 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})))))
5150adantl 481 . . . 4 ((𝑁 ∈ (ℤ‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → ((𝐹𝑗) = (𝐹𝑘) ↔ 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})))))
52 eqtr3 2631 . . . . . 6 ((𝑗 = 1 ∧ 𝑘 = 1) → 𝑗 = 𝑘)
53522a1d 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)))
54 eqid 2610 . . . . . . . . . . 11 ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
551, 54axlowdimlem13 25634 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
5655neneqd 2787 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
5756pm2.21d 117 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
5857adantrl 748 . . . . . . 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))
594, 58sylan 487 . . . . . 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))
60 iftrue 4042 . . . . . . . 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})))
61 iffalse 4045 . . . . . . . 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})))
6260, 61eqeqan12d 2626 . . . . . . 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}))))
6362imbi1d 330 . . . . . 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)))
6459, 63syl5ibr 235 . . . . 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)))
65 eqid 2610 . . . . . . . . . . . 12 ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
661, 65axlowdimlem13 25634 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
6766necomd 2837 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ≠ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
6867neneqd 2787 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ¬ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
6968pm2.21d 117 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘))
704, 69sylan 487 . . . . . . 7 ((𝑁 ∈ (ℤ‘3) ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘))
7170adantrr 749 . . . . . 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))
72 iffalse 4045 . . . . . . . 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})))
73 iftrue 4042 . . . . . . . 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})))
7472, 73eqeqan12d 2626 . . . . . . 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}))))
7574imbi1d 330 . . . . . 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)))
7671, 75syl5ibr 235 . . . . 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)))
7765, 54axlowdimlem14 25635 . . . . . . . 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))
78773expb 1258 . . . . . . 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))
794, 78sylan 487 . . . . . 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))
8072, 61eqeqan12d 2626 . . . . . . 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}))))
8180imbi1d 330 . . . . . 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)))
8279, 81syl5ibr 235 . . . . 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)))
8353, 64, 76, 824cases 987 . . . 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))
8451, 83sylbid 229 . . 3 ((𝑁 ∈ (ℤ‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → ((𝐹𝑗) = (𝐹𝑘) → 𝑗 = 𝑘))
8584ralrimivva 2954 . 2 (𝑁 ∈ (ℤ‘3) → ∀𝑗 ∈ (1...(𝑁 − 1))∀𝑘 ∈ (1...(𝑁 − 1))((𝐹𝑗) = (𝐹𝑘) → 𝑗 = 𝑘))
86 dff13 6416 . 2 (𝐹:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁) ↔ (𝐹:(1...(𝑁 − 1))⟶(𝔼‘𝑁) ∧ ∀𝑗 ∈ (1...(𝑁 − 1))∀𝑘 ∈ (1...(𝑁 − 1))((𝐹𝑗) = (𝐹𝑘) → 𝑗 = 𝑘)))
8710, 85, 86sylanbrc 695 1 (𝑁 ∈ (ℤ‘3) → 𝐹:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538  ifcif 4036  {csn 4125  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  -cneg 10146  ℕcn 10897  3c3 10948  ℤ≥cuz 11563  ...cfz 12197  𝔼cee 25568 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-ee 25571 This theorem is referenced by:  axlowdim  25641
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