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Mirrors > Home > MPE Home > Th. List > axlowdimlem9 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 28991. 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 6908 | . 2 ⊢ (𝑃‘𝐾) = (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) |
3 | eldifsn 4791 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) ↔ (𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3)) | |
4 | disjdif 4478 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
5 | 3ex 12346 | . . . . . . 7 ⊢ 3 ∈ V | |
6 | negex 11504 | . . . . . . 7 ⊢ -1 ∈ V | |
7 | 5, 6 | fnsn 6626 | . . . . . 6 ⊢ {〈3, -1〉} Fn {3} |
8 | c0ex 11253 | . . . . . . . 8 ⊢ 0 ∈ V | |
9 | 8 | fconst 6795 | . . . . . . 7 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
10 | ffn 6737 | . . . . . . 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 7001 | . . . . . 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 1450 | . . . . 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 690 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾)) |
15 | 8 | fvconst2 7224 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → ((((1...𝑁) ∖ {3}) × {0})‘𝐾) = 0) |
16 | 14, 15 | eqtrd 2775 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
17 | 3, 16 | sylbir 235 | . 2 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
18 | 2, 17 | eqtrid 2787 | 1 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 〈cop 4637 × cxp 5687 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 -cneg 11491 3c3 12320 ...cfz 13544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-neg 11493 df-2 12327 df-3 12328 |
This theorem is referenced by: axlowdimlem16 28987 axlowdimlem17 28988 |
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