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Theorem axlowdimlem9 26753
Description: Lemma for axlowdim 26764. Calculate the value of 𝑃 away from three. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypothesis
Ref Expression
axlowdimlem7.1 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
Assertion
Ref Expression
axlowdimlem9 ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃𝐾) = 0)

Proof of Theorem axlowdimlem9
StepHypRef Expression
1 axlowdimlem7.1 . . 3 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
21fveq1i 6664 . 2 (𝑃𝐾) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾)
3 eldifsn 4704 . . 3 (𝐾 ∈ ((1...𝑁) ∖ {3}) ↔ (𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3))
4 disjdif 4404 . . . . 5 ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
5 3ex 11718 . . . . . . 7 3 ∈ V
6 negex 10884 . . . . . . 7 -1 ∈ V
75, 6fnsn 6402 . . . . . 6 {⟨3, -1⟩} Fn {3}
8 c0ex 10635 . . . . . . . 8 0 ∈ V
98fconst 6557 . . . . . . 7 (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
10 ffn 6505 . . . . . . 7 ((((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} → (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}))
119, 10ax-mp 5 . . . . . 6 (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3})
12 fvun2 6748 . . . . . 6 (({⟨3, -1⟩} Fn {3} ∧ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}) ∧ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 𝐾 ∈ ((1...𝑁) ∖ {3}))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾))
137, 11, 12mp3an12 1448 . . . . 5 ((({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 𝐾 ∈ ((1...𝑁) ∖ {3})) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾))
144, 13mpan 689 . . . 4 (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾))
158fvconst2 6959 . . . 4 (𝐾 ∈ ((1...𝑁) ∖ {3}) → ((((1...𝑁) ∖ {3}) × {0})‘𝐾) = 0)
1614, 15eqtrd 2859 . . 3 (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0)
173, 16sylbir 238 . 2 ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0)
182, 17syl5eq 2871 1 ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃𝐾) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  cdif 3916  cun 3917  cin 3918  c0 4276  {csn 4550  cop 4556   × cxp 5541   Fn wfn 6340  wf 6341  cfv 6345  (class class class)co 7151  0cc0 10537  1c1 10538  -cneg 10871  3c3 11692  ...cfz 12896
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-mulcl 10599  ax-i2m1 10605
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-fv 6353  df-ov 7154  df-neg 10873  df-2 11699  df-3 11700
This theorem is referenced by:  axlowdimlem16  26760  axlowdimlem17  26761
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