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Theorem axlowdimlem15 28481
Description: Lemma for axlowdim 28486. 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 2730 . . . . . 6 ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
21axlowdimlem7 28473 . . . . 5 (𝑁 ∈ (ℤ‘3) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
32adantr 479 . . . 4 ((𝑁 ∈ (ℤ‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁))
4 eluzge3nn 12878 . . . . 5 (𝑁 ∈ (ℤ‘3) → 𝑁 ∈ ℕ)
5 eqid 2730 . . . . . 6 ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))
65axlowdimlem10 28476 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
74, 6sylan 578 . . . 4 ((𝑁 ∈ (ℤ‘3) ∧ 𝑖 ∈ (1...(𝑁 − 1))) → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) ∈ (𝔼‘𝑁))
83, 7ifcld 4573 . . 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 7114 . 2 (𝑁 ∈ (ℤ‘3) → 𝐹:(1...(𝑁 − 1))⟶(𝔼‘𝑁))
11 eqeq1 2734 . . . . . . . 8 (𝑖 = 𝑗 → (𝑖 = 1 ↔ 𝑗 = 1))
12 oveq1 7418 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
1312opeq1d 4878 . . . . . . . . . 10 (𝑖 = 𝑗 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑗 + 1), 1⟩)
1413sneqd 4639 . . . . . . . . 9 (𝑖 = 𝑗 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑗 + 1), 1⟩})
1512sneqd 4639 . . . . . . . . . . 11 (𝑖 = 𝑗 → {(𝑖 + 1)} = {(𝑗 + 1)})
1615difeq2d 4121 . . . . . . . . . 10 (𝑖 = 𝑗 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑗 + 1)}))
1716xpeq1d 5704 . . . . . . . . 9 (𝑖 = 𝑗 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
1814, 17uneq12d 4163 . . . . . . . 8 (𝑖 = 𝑗 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
1911, 18ifbieq2d 4553 . . . . . . 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 5430 . . . . . . . . 9 {⟨3, -1⟩} ∈ V
21 ovex 7444 . . . . . . . . . . 11 (1...𝑁) ∈ V
2221difexi 5327 . . . . . . . . . 10 ((1...𝑁) ∖ {3}) ∈ V
23 snex 5430 . . . . . . . . . 10 {0} ∈ V
2422, 23xpex 7742 . . . . . . . . 9 (((1...𝑁) ∖ {3}) × {0}) ∈ V
2520, 24unex 7735 . . . . . . . 8 ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ V
26 snex 5430 . . . . . . . . 9 {⟨(𝑗 + 1), 1⟩} ∈ V
2721difexi 5327 . . . . . . . . . 10 ((1...𝑁) ∖ {(𝑗 + 1)}) ∈ V
2827, 23xpex 7742 . . . . . . . . 9 (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}) ∈ V
2926, 28unex 7735 . . . . . . . 8 ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ∈ V
3025, 29ifex 4577 . . . . . . 7 if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))) ∈ V
3119, 9, 30fvmpt 6997 . . . . . 6 (𝑗 ∈ (1...(𝑁 − 1)) → (𝐹𝑗) = if(𝑗 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))))
32 eqeq1 2734 . . . . . . . 8 (𝑖 = 𝑘 → (𝑖 = 1 ↔ 𝑘 = 1))
33 oveq1 7418 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
3433opeq1d 4878 . . . . . . . . . 10 (𝑖 = 𝑘 → ⟨(𝑖 + 1), 1⟩ = ⟨(𝑘 + 1), 1⟩)
3534sneqd 4639 . . . . . . . . 9 (𝑖 = 𝑘 → {⟨(𝑖 + 1), 1⟩} = {⟨(𝑘 + 1), 1⟩})
3633sneqd 4639 . . . . . . . . . . 11 (𝑖 = 𝑘 → {(𝑖 + 1)} = {(𝑘 + 1)})
3736difeq2d 4121 . . . . . . . . . 10 (𝑖 = 𝑘 → ((1...𝑁) ∖ {(𝑖 + 1)}) = ((1...𝑁) ∖ {(𝑘 + 1)}))
3837xpeq1d 5704 . . . . . . . . 9 (𝑖 = 𝑘 → (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}) = (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
3935, 38uneq12d 4163 . . . . . . . 8 (𝑖 = 𝑘 → ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
4032, 39ifbieq2d 4553 . . . . . . 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 5430 . . . . . . . . 9 {⟨(𝑘 + 1), 1⟩} ∈ V
4221difexi 5327 . . . . . . . . . 10 ((1...𝑁) ∖ {(𝑘 + 1)}) ∈ V
4342, 23xpex 7742 . . . . . . . . 9 (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}) ∈ V
4441, 43unex 7735 . . . . . . . 8 ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) ∈ V
4525, 44ifex 4577 . . . . . . 7 if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))) ∈ V
4640, 9, 45fvmpt 6997 . . . . . 6 (𝑘 ∈ (1...(𝑁 − 1)) → (𝐹𝑘) = if(𝑘 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))))
4731, 46eqeqan12d 2744 . . . . 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})))))
4847adantl 480 . . . 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})))))
49 eqtr3 2756 . . . . . 6 ((𝑗 = 1 ∧ 𝑘 = 1) → 𝑗 = 𝑘)
50492a1d 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 2730 . . . . . . . . . . 11 ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0}))
521, 51axlowdimlem13 28479 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
5352neneqd 2943 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})))
5453pm2.21d 121 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝑘 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑘 + 1)}) × {0})) → 𝑗 = 𝑘))
5554adantrl 712 . . . . . . 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))
564, 55sylan 578 . . . . . 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 4533 . . . . . . . 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 4536 . . . . . . . 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})))
5957, 58eqeqan12d 2744 . . . . . . 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}))))
6059imbi1d 340 . . . . . 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)))
6156, 60imbitrrid 245 . . . . 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 2730 . . . . . . . . . . . 12 ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0}))
631, 62axlowdimlem13 28479 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) ≠ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})))
6463necomd 2994 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) ≠ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
6564neneqd 2943 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ¬ ({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})))
6665pm2.21d 121 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘))
674, 66sylan 578 . . . . . . 7 ((𝑁 ∈ (ℤ‘3) ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (({⟨(𝑗 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑗 + 1)}) × {0})) = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) → 𝑗 = 𝑘))
6867adantrr 713 . . . . . 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 4536 . . . . . . . 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 4533 . . . . . . . 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})))
7169, 70eqeqan12d 2744 . . . . . . 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}))))
7271imbi1d 340 . . . . . 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)))
7368, 72imbitrrid 245 . . . . 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)))
7462, 51axlowdimlem14 28480 . . . . . . . 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))
75743expb 1118 . . . . . . 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))
764, 75sylan 578 . . . . . 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))
7769, 58eqeqan12d 2744 . . . . . . 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}))))
7877imbi1d 340 . . . . . 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)))
7976, 78imbitrrid 245 . . . . 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)))
8050, 61, 73, 794cases 1037 . . . 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))
8148, 80sylbid 239 . . 3 ((𝑁 ∈ (ℤ‘3) ∧ (𝑗 ∈ (1...(𝑁 − 1)) ∧ 𝑘 ∈ (1...(𝑁 − 1)))) → ((𝐹𝑗) = (𝐹𝑘) → 𝑗 = 𝑘))
8281ralrimivva 3198 . 2 (𝑁 ∈ (ℤ‘3) → ∀𝑗 ∈ (1...(𝑁 − 1))∀𝑘 ∈ (1...(𝑁 − 1))((𝐹𝑗) = (𝐹𝑘) → 𝑗 = 𝑘))
83 dff13 7256 . 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)))
8410, 82, 83sylanbrc 581 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 205  wa 394   = wceq 1539  wcel 2104  wral 3059  cdif 3944  cun 3945  ifcif 4527  {csn 4627  cop 4633  cmpt 5230   × cxp 5673  wf 6538  1-1wf1 6539  cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  cmin 11448  -cneg 11449  cn 12216  3c3 12272  cuz 12826  ...cfz 13488  𝔼cee 28413
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-ee 28416
This theorem is referenced by:  axlowdim  28486
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