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Mirrors > Home > MPE Home > Th. List > axlowdimlem8 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26749. 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 6673 | . 2 ⊢ (𝑃‘3) = (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) |
3 | 3ex 11722 | . . . 4 ⊢ 3 ∈ V | |
4 | negex 10886 | . . . 4 ⊢ -1 ∈ V | |
5 | 3, 4 | fnsn 6414 | . . 3 ⊢ {〈3, -1〉} Fn {3} |
6 | c0ex 10637 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6567 | . . . 4 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} |
8 | ffn 6516 | . . . 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 4423 | . . . 4 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
11 | 3 | snid 4603 | . . . 4 ⊢ 3 ∈ {3} |
12 | 10, 11 | pm3.2i 473 | . . 3 ⊢ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3}) |
13 | fvun1 6756 | . . 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 1457 | . 2 ⊢ (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({〈3, -1〉}‘3) |
15 | 3, 4 | fvsn 6945 | . 2 ⊢ ({〈3, -1〉}‘3) = -1 |
16 | 2, 14, 15 | 3eqtri 2850 | 1 ⊢ (𝑃‘3) = -1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 〈cop 4575 × cxp 5555 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 -cneg 10873 3c3 11696 ...cfz 12895 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-mulcl 10601 ax-i2m1 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-neg 10875 df-2 11703 df-3 11704 |
This theorem is referenced by: axlowdimlem16 26745 |
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