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Mirrors > Home > MPE Home > Th. List > axlowdimlem8 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26674. Calculate the value of 𝑃 at three. (Contributed by Scott Fenton, 21-Apr-2013.) |
Ref | Expression |
---|---|
axlowdimlem7.1 | ⊢ 𝑃 = ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) |
Ref | Expression |
---|---|
axlowdimlem8 | ⊢ (𝑃‘3) = -1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axlowdimlem7.1 | . . 3 ⊢ 𝑃 = ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) | |
2 | 1 | fveq1i 6664 | . 2 ⊢ (𝑃‘3) = (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) |
3 | 3ex 11707 | . . . 4 ⊢ 3 ∈ V | |
4 | negex 10872 | . . . 4 ⊢ -1 ∈ V | |
5 | 3, 4 | fnsn 6405 | . . 3 ⊢ {〈3, -1〉} Fn {3} |
6 | c0ex 10623 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6558 | . . . 4 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
8 | ffn 6507 | . . . 4 ⊢ ((((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} → (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3})) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}) |
10 | disjdif 4417 | . . . 4 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
11 | 3 | snid 4591 | . . . 4 ⊢ 3 ∈ {3} |
12 | 10, 11 | pm3.2i 471 | . . 3 ⊢ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3}) |
13 | fvun1 6747 | . . 3 ⊢ (({〈3, -1〉} Fn {3} ∧ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}) ∧ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({〈3, -1〉}‘3)) | |
14 | 5, 9, 12, 13 | mp3an 1452 | . 2 ⊢ (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({〈3, -1〉}‘3) |
15 | 3, 4 | fvsn 6935 | . 2 ⊢ ({〈3, -1〉}‘3) = -1 |
16 | 2, 14, 15 | 3eqtri 2845 | 1 ⊢ (𝑃‘3) = -1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 {csn 4557 〈cop 4563 × cxp 5546 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 -cneg 10859 3c3 11681 ...cfz 12880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-neg 10861 df-2 11688 df-3 11689 |
This theorem is referenced by: axlowdimlem16 26670 |
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