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Theorem axlowdimlem8 28780

Description: Lemma for axlowdim 28792. 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**
Step	Hyp	Ref	Expression
1		axlowdimlem7.1	. . 3 ⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
2	1	fveq1i 6903	. 2 ⊢ (𝑃‘3) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3)
3		3ex 12332	. . . 4 ⊢ 3 ∈ V
4		negex 11496	. . . 4 ⊢ -1 ∈ V
5	3, 4	fnsn 6616	. . 3 ⊢ {⟨3, -1⟩} Fn {3}
6		c0ex 11246	. . . . 5 ⊢ 0 ∈ V
7	6	fconst 6788	. . . 4 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
8		ffn 6727	. . . 4 ⊢ ((((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} → (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}))
9	7, 8	ax-mp 5	. . 3 ⊢ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3})
10		disjdif 4475	. . . 4 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
11	3	snid 4669	. . . 4 ⊢ 3 ∈ {3}
12	10, 11	pm3.2i 469	. . 3 ⊢ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})
13		fvun1 6994	. . 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))
14	5, 9, 12, 13	mp3an 1457	. 2 ⊢ (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3)
15	3, 4	fvsn 7196	. 2 ⊢ ({⟨3, -1⟩}‘3) = -1
16	2, 14, 15	3eqtri 2760	1 ⊢ (𝑃‘3) = -1

Colors of variables: wff setvar class

Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ∅c0 4326 {csn 4632 ⟨cop 4638 × cxp 5680 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 -cneg 11483 3c3 12306 ...cfz 13524

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-mulcl 11208 ax-i2m1 11214

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-neg 11485 df-2 12313 df-3 12314

This theorem is referenced by: axlowdimlem16 28788

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