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Mirrors > Home > MPE Home > Th. List > axlowdimlem11 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 27739. Calculate the value of 𝑄 at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.) |
Ref | Expression |
---|---|
axlowdimlem10.1 | ⊢ 𝑄 = ({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) |
Ref | Expression |
---|---|
axlowdimlem11 | ⊢ (𝑄‘(𝐼 + 1)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axlowdimlem10.1 | . . 3 ⊢ 𝑄 = ({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) | |
2 | 1 | fveq1i 6840 | . 2 ⊢ (𝑄‘(𝐼 + 1)) = (({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) |
3 | ovex 7384 | . . . 4 ⊢ (𝐼 + 1) ∈ V | |
4 | 1ex 11109 | . . . 4 ⊢ 1 ∈ V | |
5 | 3, 4 | fnsn 6556 | . . 3 ⊢ {〈(𝐼 + 1), 1〉} Fn {(𝐼 + 1)} |
6 | c0ex 11107 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6725 | . . . 4 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0} |
8 | ffn 6665 | . . . 4 ⊢ ((((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0} → (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}) Fn ((1...𝑁) ∖ {(𝐼 + 1)})) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}) Fn ((1...𝑁) ∖ {(𝐼 + 1)}) |
10 | disjdif 4429 | . . . 4 ⊢ ({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ | |
11 | 3 | snid 4620 | . . . 4 ⊢ (𝐼 + 1) ∈ {(𝐼 + 1)} |
12 | 10, 11 | pm3.2i 471 | . . 3 ⊢ (({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ ∧ (𝐼 + 1) ∈ {(𝐼 + 1)}) |
13 | fvun1 6929 | . . 3 ⊢ (({〈(𝐼 + 1), 1〉} Fn {(𝐼 + 1)} ∧ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}) Fn ((1...𝑁) ∖ {(𝐼 + 1)}) ∧ (({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ ∧ (𝐼 + 1) ∈ {(𝐼 + 1)})) → (({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) = ({〈(𝐼 + 1), 1〉}‘(𝐼 + 1))) | |
14 | 5, 9, 12, 13 | mp3an 1461 | . 2 ⊢ (({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) = ({〈(𝐼 + 1), 1〉}‘(𝐼 + 1)) |
15 | 3, 4 | fvsn 7123 | . 2 ⊢ ({〈(𝐼 + 1), 1〉}‘(𝐼 + 1)) = 1 |
16 | 2, 14, 15 | 3eqtri 2769 | 1 ⊢ (𝑄‘(𝐼 + 1)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3905 ∪ cun 3906 ∩ cin 3907 ∅c0 4280 {csn 4584 〈cop 4590 × cxp 5629 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 0cc0 11009 1c1 11010 + caddc 11012 ...cfz 13378 |
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 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-mulcl 11071 ax-i2m1 11077 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 |
This theorem is referenced by: axlowdimlem14 27733 axlowdimlem16 27735 |
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