![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axlowdimlem8 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 28219. 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 6893 | . 2 ⊢ (𝑃‘3) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) |
3 | 3ex 12294 | . . . 4 ⊢ 3 ∈ V | |
4 | negex 11458 | . . . 4 ⊢ -1 ∈ V | |
5 | 3, 4 | fnsn 6607 | . . 3 ⊢ {⟨3, -1⟩} Fn {3} |
6 | c0ex 11208 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6778 | . . . 4 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
8 | ffn 6718 | . . . 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 4472 | . . . 4 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
11 | 3 | snid 4665 | . . . 4 ⊢ 3 ∈ {3} |
12 | 10, 11 | pm3.2i 472 | . . 3 ⊢ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3}) |
13 | fvun1 6983 | . . 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 1462 | . 2 ⊢ (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3) |
15 | 3, 4 | fvsn 7179 | . 2 ⊢ ({⟨3, -1⟩}‘3) = -1 |
16 | 2, 14, 15 | 3eqtri 2765 | 1 ⊢ (𝑃‘3) = -1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ 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 7409 0cc0 11110 1c1 11111 -cneg 11445 3c3 12268 ...cfz 13484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-mulcl 11172 ax-i2m1 11178 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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 7412 df-neg 11447 df-2 12275 df-3 12276 |
This theorem is referenced by: axlowdimlem16 28215 |
Copyright terms: Public domain | W3C validator |