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Mirrors > Home > MPE Home > Th. List > axlowdimlem9 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 28483. 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 6893 | . 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 4472 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
5 | 3ex 12299 | . . . . . . 7 ⊢ 3 ∈ V | |
6 | negex 11463 | . . . . . . 7 ⊢ -1 ∈ V | |
7 | 5, 6 | fnsn 6607 | . . . . . 6 ⊢ {⟨3, -1⟩} Fn {3} |
8 | c0ex 11213 | . . . . . . . 8 ⊢ 0 ∈ V | |
9 | 8 | fconst 6778 | . . . . . . 7 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
10 | ffn 6718 | . . . . . . 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 6984 | . . . . . 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 687 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾)) |
15 | 8 | fvconst2 7208 | . . . 4 ⊢ (𝐾 ∈ ((1...𝑁) ∖ {3}) → ((((1...𝑁) ∖ {3}) × {0})‘𝐾) = 0) |
16 | 14, 15 | eqtrd 2771 | . . 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 2783 | 1 ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ∅c0 4323 {csn 4629 ⟨cop 4635 × cxp 5675 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 0cc0 11113 1c1 11114 -cneg 11450 3c3 12273 ...cfz 13489 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-mulcl 11175 ax-i2m1 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7415 df-neg 11452 df-2 12280 df-3 12281 |
This theorem is referenced by: axlowdimlem16 28479 axlowdimlem17 28480 |
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