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Mirrors > Home > MPE Home > Th. List > axlowdimlem11 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 28759. 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 6892 | . 2 ⊢ (𝑄‘(𝐼 + 1)) = (({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) |
3 | ovex 7447 | . . . 4 ⊢ (𝐼 + 1) ∈ V | |
4 | 1ex 11232 | . . . 4 ⊢ 1 ∈ V | |
5 | 3, 4 | fnsn 6605 | . . 3 ⊢ {⟨(𝐼 + 1), 1⟩} Fn {(𝐼 + 1)} |
6 | c0ex 11230 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | fconst 6777 | . . . 4 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0} |
8 | ffn 6716 | . . . 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 4467 | . . . 4 ⊢ ({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ | |
11 | 3 | snid 4660 | . . . 4 ⊢ (𝐼 + 1) ∈ {(𝐼 + 1)} |
12 | 10, 11 | pm3.2i 470 | . . 3 ⊢ (({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ ∧ (𝐼 + 1) ∈ {(𝐼 + 1)}) |
13 | fvun1 6983 | . . 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 1458 | . 2 ⊢ (({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) = ({⟨(𝐼 + 1), 1⟩}‘(𝐼 + 1)) |
15 | 3, 4 | fvsn 7184 | . 2 ⊢ ({⟨(𝐼 + 1), 1⟩}‘(𝐼 + 1)) = 1 |
16 | 2, 14, 15 | 3eqtri 2759 | 1 ⊢ (𝑄‘(𝐼 + 1)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ∅c0 4318 {csn 4624 ⟨cop 4630 × cxp 5670 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 ...cfz 13508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-mulcl 11192 ax-i2m1 11198 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 |
This theorem is referenced by: axlowdimlem14 28753 axlowdimlem16 28755 |
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