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Mirrors > Home > MPE Home > Th. List > axlowdimlem9 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26749. 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 6673 | . 2 ⊢ (𝑃‘𝐾) = (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) |
3 | eldifsn 4721 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) ↔ (𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3)) | |
4 | disjdif 4423 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
5 | 3ex 11722 | . . . . . . 7 ⊢ 3 ∈ V | |
6 | negex 10886 | . . . . . . 7 ⊢ -1 ∈ V | |
7 | 5, 6 | fnsn 6414 | . . . . . 6 ⊢ {〈3, -1〉} Fn {3} |
8 | c0ex 10637 | . . . . . . . 8 ⊢ 0 ∈ V | |
9 | 8 | fconst 6567 | . . . . . . 7 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
10 | ffn 6516 | . . . . . . 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 6757 | . . . . . 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 1447 | . . . . 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 688 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾)) |
15 | 8 | fvconst2 6968 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → ((((1...𝑁) ∖ {3}) × {0})‘𝐾) = 0) |
16 | 14, 15 | eqtrd 2858 | . . 3 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
17 | 3, 16 | sylbir 237 | . 2 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0) |
18 | 2, 17 | syl5eq 2870 | 1 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 〈cop 4575 × cxp 5555 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 -cneg 10873 3c3 11696 ...cfz 12895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-mulcl 10601 ax-i2m1 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-neg 10875 df-2 11703 df-3 11704 |
This theorem is referenced by: axlowdimlem16 26745 axlowdimlem17 26746 |
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