![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axlowdimlem8 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 27908. 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 6843 | . 2 ⊢ (𝑃‘3) = (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) |
3 | 3ex 12234 | . . . 4 ⊢ 3 ∈ V | |
4 | negex 11398 | . . . 4 ⊢ -1 ∈ V | |
5 | 3, 4 | fnsn 6559 | . . 3 ⊢ {〈3, -1〉} Fn {3} |
6 | c0ex 11148 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6728 | . . . 4 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
8 | ffn 6668 | . . . 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 4431 | . . . 4 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
11 | 3 | snid 4622 | . . . 4 ⊢ 3 ∈ {3} |
12 | 10, 11 | pm3.2i 471 | . . 3 ⊢ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3}) |
13 | fvun1 6932 | . . 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 1461 | . 2 ⊢ (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({〈3, -1〉}‘3) |
15 | 3, 4 | fvsn 7126 | . 2 ⊢ ({〈3, -1〉}‘3) = -1 |
16 | 2, 14, 15 | 3eqtri 2768 | 1 ⊢ (𝑃‘3) = -1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ∅c0 4282 {csn 4586 〈cop 4592 × cxp 5631 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 0cc0 11050 1c1 11051 -cneg 11385 3c3 12208 ...cfz 13423 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-mulcl 11112 ax-i2m1 11118 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-ov 7359 df-neg 11387 df-2 12215 df-3 12216 |
This theorem is referenced by: axlowdimlem16 27904 |
Copyright terms: Public domain | W3C validator |