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Mirrors > Home > MPE Home > Th. List > axlowdimlem9 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26755. 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 6646 | . 2 ⊢ (𝑃‘𝐾) = (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) |
3 | eldifsn 4680 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) ↔ (𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3)) | |
4 | disjdif 4379 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
5 | 3ex 11707 | . . . . . . 7 ⊢ 3 ∈ V | |
6 | negex 10873 | . . . . . . 7 ⊢ -1 ∈ V | |
7 | 5, 6 | fnsn 6382 | . . . . . 6 ⊢ {〈3, -1〉} Fn {3} |
8 | c0ex 10624 | . . . . . . . 8 ⊢ 0 ∈ V | |
9 | 8 | fconst 6539 | . . . . . . 7 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
10 | ffn 6487 | . . . . . . 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 6730 | . . . . . 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 1448 | . . . . 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 689 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾)) |
15 | 8 | fvconst2 6943 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → ((((1...𝑁) ∖ {3}) × {0})‘𝐾) = 0) |
16 | 14, 15 | eqtrd 2833 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
17 | 3, 16 | sylbir 238 | . 2 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
18 | 2, 17 | syl5eq 2845 | 1 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 〈cop 4531 × cxp 5517 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 -cneg 10860 3c3 11681 ...cfz 12885 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-neg 10862 df-2 11688 df-3 11689 |
This theorem is referenced by: axlowdimlem16 26751 axlowdimlem17 26752 |
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