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Theorem axlowdimlem15 25881
Description: Lemma for axlowdim 25886. 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 2651 . . . . . 6 ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
21axlowdimlem7 25873 . . . . 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 11768 . . . . 5 (𝑁 ∈ (ℤ‘3) → 𝑁 ∈ ℕ)
5 eqid 2651 . . . . . 6 ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))
65axlowdimlem10 25876 . . . . 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 4164 . . 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 6425 . 2 (𝑁 ∈ (ℤ‘3) → 𝐹:(1...(𝑁 − 1))⟶(𝔼‘𝑁))
11 eqeq1 2655 . . . . . . . 8 (𝑖 = 𝑗 → (𝑖 = 1 ↔ 𝑗 = 1))
12 oveq1 6697 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
1312opeq1d 4439 . . . . . . . . . 10 (𝑖 = 𝑗 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑗 + 1), 1⟩)
1413sneqd 4222 . . . . . . . . 9 (𝑖 = 𝑗 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑗 + 1), 1⟩})
1512sneqd 4222 . . . . . . . . . . 11 (𝑖 = 𝑗 → {(𝑖 + 1)} = {(𝑗 + 1)})
1615difeq2d 3761 . . . . . . . . . 10 (𝑖 = 𝑗 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑗 + 1)}))
1716xpeq1d 5172 . . . . . . . . 9 (𝑖 = 𝑗 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
1814, 17uneq12d 3801 . . . . . . . 8 (𝑖 = 𝑗 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
1911, 18ifbieq2d 4144 . . . . . . 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 4938 . . . . . . . . 9 {⟨3, -1⟩} ∈ V
21 ovex 6718 . . . . . . . . . . 11 (1...𝑁) ∈ V
22 difexg 4841 . . . . . . . . . . 11 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {3}) ∈ V)
2321, 22ax-mp 5 . . . . . . . . . 10 ((1...𝑁) ∖ {3}) ∈ V
24 snex 4938 . . . . . . . . . 10 {0} ∈ V
2523, 24xpex 7004 . . . . . . . . 9 (((1...𝑁) ∖ {3}) × {0}) ∈ V
2620, 25unex 6998 . . . . . . . 8 ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ V
27 snex 4938 . . . . . . . . 9 {⟨(𝑗 + 1), 1⟩} ∈ V
28 difexg 4841 . . . . . . . . . . 11 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {(𝑗 + 1)}) ∈ V)
2921, 28ax-mp 5 . . . . . . . . . 10 ((1...𝑁) ∖ {(𝑗 + 1)}) ∈ V
3029, 24xpex 7004 . . . . . . . . 9 (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}) ∈ V
3127, 30unex 6998 . . . . . . . 8 ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ∈ V
3226, 31ifex 4189 . . . . . . 7 if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) ∈ V
3319, 9, 32fvmpt 6321 . . . . . 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 2655 . . . . . . . 8 (𝑖 = 𝑘 → (𝑖 = 1 ↔ 𝑘 = 1))
35 oveq1 6697 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
3635opeq1d 4439 . . . . . . . . . 10 (𝑖 = 𝑘 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑘 + 1), 1⟩)
3736sneqd 4222 . . . . . . . . 9 (𝑖 = 𝑘 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑘 + 1), 1⟩})
3835sneqd 4222 . . . . . . . . . . 11 (𝑖 = 𝑘 → {(𝑖 + 1)} = {(𝑘 + 1)})
3938difeq2d 3761 . . . . . . . . . 10 (𝑖 = 𝑘 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑘 + 1)}))
4039xpeq1d 5172 . . . . . . . . 9 (𝑖 = 𝑘 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
4137, 40uneq12d 3801 . . . . . . . 8 (𝑖 = 𝑘 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
4234, 41ifbieq2d 4144 . . . . . . 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 4938 . . . . . . . . 9 {⟨(𝑘 + 1), 1⟩} ∈ V
44 difexg 4841 . . . . . . . . . . 11 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {(𝑘 + 1)}) ∈ V)
4521, 44ax-mp 5 . . . . . . . . . 10 ((1...𝑁) ∖ {(𝑘 + 1)}) ∈ V
4645, 24xpex 7004 . . . . . . . . 9 (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}) ∈ V
4743, 46unex 6998 . . . . . . . 8 ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) ∈ V
4826, 47ifex 4189 . . . . . . 7 if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ∈ V
4942, 9, 48fvmpt 6321 . . . . . 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 2667 . . . . 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 2672 . . . . . 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 2651 . . . . . . . . . . 11 ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
551, 54axlowdimlem13 25879 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
5655neneqd 2828 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
5756pm2.21d 118 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
5857adantrl 752 . . . . . . 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 4125 . . . . . . . 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 4128 . . . . . . . 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 2667 . . . . . . 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 236 . . . . 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 2651 . . . . . . . . . . . 12 ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
661, 65axlowdimlem13 25879 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
6766necomd 2878 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ≠ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
6867neneqd 2828 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ¬ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
6968pm2.21d 118 . . . . . . . 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 753 . . . . . 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 4128 . . . . . . . 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 4125 . . . . . . . 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 2667 . . . . . . 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 236 . . . . 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 25880 . . . . . . . 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 1285 . . . . . . 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 2667 . . . . . . 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 236 . . . . 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 1009 . . . 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 230 . . 3 ((𝑁 ∈ (ℤ‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → ((𝐹𝑗) = (𝐹𝑘) → 𝑗 = 𝑘))
8584ralrimivva 3000 . 2 (𝑁 ∈ (ℤ‘3) → ∀𝑗 ∈ (1...(𝑁 − 1))∀𝑘 ∈ (1...(𝑁 − 1))((𝐹𝑗) = (𝐹𝑘) → 𝑗 = 𝑘))
86 dff13 6552 . 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 699 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  cun 3605  ifcif 4119  {csn 4210  cop 4216  cmpt 4762   × cxp 5141  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cmin 10304  -cneg 10305  cn 11058  3c3 11109  cuz 11725  ...cfz 12364  𝔼cee 25813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-ee 25816
This theorem is referenced by:  axlowdim  25886
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