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Theorem axlowdimlem8 25746
Description: Lemma for axlowdim 25758. Calculate the value of 𝑃 at three. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypothesis
Ref Expression
axlowdimlem7.1 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
Assertion
Ref Expression
axlowdimlem8 (𝑃‘3) = -1

Proof of Theorem axlowdimlem8
StepHypRef Expression
1 axlowdimlem7.1 . . 3 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
21fveq1i 6154 . 2 (𝑃‘3) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3)
3 3re 11046 . . . . 5 3 ∈ ℝ
43elexi 3202 . . . 4 3 ∈ V
5 negex 10231 . . . 4 -1 ∈ V
64, 5fnsn 5909 . . 3 {⟨3, -1⟩} Fn {3}
7 c0ex 9986 . . . . 5 0 ∈ V
87fconst 6053 . . . 4 (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
9 ffn 6007 . . . 4 ((((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} → (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}))
108, 9ax-mp 5 . . 3 (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3})
11 disjdif 4017 . . . 4 ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
124snid 4184 . . . 4 3 ∈ {3}
1311, 12pm3.2i 471 . . 3 (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})
14 fvun1 6231 . . 3 (({⟨3, -1⟩} Fn {3} ∧ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}) ∧ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3))
156, 10, 13, 14mp3an 1421 . 2 (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3)
164, 5fvsn 6406 . 2 ({⟨3, -1⟩}‘3) = -1
172, 15, 163eqtri 2647 1 (𝑃‘3) = -1
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  cdif 3556  cun 3557  cin 3558  c0 3896  {csn 4153  cop 4159   × cxp 5077   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  cr 9887  0cc0 9888  1c1 9889  -cneg 10219  3c3 11023  ...cfz 12276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-i2m1 9956  ax-1ne0 9957  ax-rrecex 9960  ax-cnre 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-neg 10221  df-2 11031  df-3 11032
This theorem is referenced by:  axlowdimlem16  25754
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