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Mirrors > Home > MPE Home > Th. List > axlowdimlem8 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 27024. 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 6707 | . 2 ⊢ (𝑃‘3) = (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) |
3 | 3ex 11895 | . . . 4 ⊢ 3 ∈ V | |
4 | negex 11059 | . . . 4 ⊢ -1 ∈ V | |
5 | 3, 4 | fnsn 6427 | . . 3 ⊢ {〈3, -1〉} Fn {3} |
6 | c0ex 10810 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6594 | . . . 4 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
8 | ffn 6534 | . . . 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 4376 | . . . 4 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
11 | 3 | snid 4567 | . . . 4 ⊢ 3 ∈ {3} |
12 | 10, 11 | pm3.2i 474 | . . 3 ⊢ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3}) |
13 | fvun1 6791 | . . 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 1463 | . 2 ⊢ (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({〈3, -1〉}‘3) |
15 | 3, 4 | fvsn 6985 | . 2 ⊢ ({〈3, -1〉}‘3) = -1 |
16 | 2, 14, 15 | 3eqtri 2766 | 1 ⊢ (𝑃‘3) = -1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∖ cdif 3854 ∪ cun 3855 ∩ cin 3856 ∅c0 4227 {csn 4531 〈cop 4537 × cxp 5538 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 0cc0 10712 1c1 10713 -cneg 11046 3c3 11869 ...cfz 13078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-mulcl 10774 ax-i2m1 10780 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-neg 11048 df-2 11876 df-3 11877 |
This theorem is referenced by: axlowdimlem16 27020 |
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