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Mirrors > Home > MPE Home > Th. List > axlowdimlem11 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 25841. 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 6192 | . 2 ⊢ (𝑄‘(𝐼 + 1)) = (({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) |
3 | ovex 6678 | . . . 4 ⊢ (𝐼 + 1) ∈ V | |
4 | 1ex 10035 | . . . 4 ⊢ 1 ∈ V | |
5 | 3, 4 | fnsn 5946 | . . 3 ⊢ {〈(𝐼 + 1), 1〉} Fn {(𝐼 + 1)} |
6 | c0ex 10034 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6091 | . . . 4 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0} |
8 | ffn 6045 | . . . 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 4040 | . . . 4 ⊢ ({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ | |
11 | 3 | snid 4208 | . . . 4 ⊢ (𝐼 + 1) ∈ {(𝐼 + 1)} |
12 | 10, 11 | pm3.2i 471 | . . 3 ⊢ (({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ ∧ (𝐼 + 1) ∈ {(𝐼 + 1)}) |
13 | fvun1 6269 | . . 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 1424 | . 2 ⊢ (({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) = ({〈(𝐼 + 1), 1〉}‘(𝐼 + 1)) |
15 | 3, 4 | fvsn 6446 | . 2 ⊢ ({〈(𝐼 + 1), 1〉}‘(𝐼 + 1)) = 1 |
16 | 2, 14, 15 | 3eqtri 2648 | 1 ⊢ (𝑄‘(𝐼 + 1)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 {csn 4177 〈cop 4183 × cxp 5112 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 ...cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 |
This theorem is referenced by: axlowdimlem14 25835 axlowdimlem16 25837 |
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