![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axlowdimlem9 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 28486. Calculate the value of 𝑃 away from three. (Contributed by Scott Fenton, 21-Apr-2013.) |
Ref | Expression |
---|---|
axlowdimlem7.1 | ⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) |
Ref | Expression |
---|---|
axlowdimlem9 | ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃‘𝐾) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axlowdimlem7.1 | . . 3 ⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) | |
2 | 1 | fveq1i 6891 | . 2 ⊢ (𝑃‘𝐾) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) |
3 | eldifsn 4789 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) ↔ (𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3)) | |
4 | disjdif 4470 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
5 | 3ex 12298 | . . . . . . 7 ⊢ 3 ∈ V | |
6 | negex 11462 | . . . . . . 7 ⊢ -1 ∈ V | |
7 | 5, 6 | fnsn 6605 | . . . . . 6 ⊢ {⟨3, -1⟩} Fn {3} |
8 | c0ex 11212 | . . . . . . . 8 ⊢ 0 ∈ V | |
9 | 8 | fconst 6776 | . . . . . . 7 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
10 | ffn 6716 | . . . . . . 7 ⊢ ((((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} → (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3})) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}) |
12 | fvun2 6982 | . . . . . 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)) | |
13 | 7, 11, 12 | mp3an12 1449 | . . . . 5 ⊢ ((({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 𝐾 ∈ ((1...𝑁) ∖ {3})) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾)) |
14 | 4, 13 | mpan 686 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾)) |
15 | 8 | fvconst2 7206 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → ((((1...𝑁) ∖ {3}) × {0})‘𝐾) = 0) |
16 | 14, 15 | eqtrd 2770 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
17 | 3, 16 | sylbir 234 | . 2 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
18 | 2, 17 | eqtrid 2782 | 1 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 {csn 4627 ⟨cop 4633 × cxp 5673 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 0cc0 11112 1c1 11113 -cneg 11449 3c3 12272 ...cfz 13488 |
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-pr 5426 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-mulcl 11174 ax-i2m1 11180 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 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-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7414 df-neg 11451 df-2 12279 df-3 12280 |
This theorem is referenced by: axlowdimlem16 28482 axlowdimlem17 28483 |
Copyright terms: Public domain | W3C validator |