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| Mirrors > Home > MPE Home > Th. List > axlowdimlem11 | Structured version Visualization version GIF version | ||
| Description: Lemma for axlowdim 28976. 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 6907 | . 2 ⊢ (𝑄‘(𝐼 + 1)) = (({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) |
| 3 | ovex 7464 | . . . 4 ⊢ (𝐼 + 1) ∈ V | |
| 4 | 1ex 11257 | . . . 4 ⊢ 1 ∈ V | |
| 5 | 3, 4 | fnsn 6624 | . . 3 ⊢ {〈(𝐼 + 1), 1〉} Fn {(𝐼 + 1)} |
| 6 | c0ex 11255 | . . . . 5 ⊢ 0 ∈ V | |
| 7 | 6 | fconst 6794 | . . . 4 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0} |
| 8 | ffn 6736 | . . . 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 4472 | . . . 4 ⊢ ({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ | |
| 11 | 3 | snid 4662 | . . . 4 ⊢ (𝐼 + 1) ∈ {(𝐼 + 1)} |
| 12 | 10, 11 | pm3.2i 470 | . . 3 ⊢ (({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ ∧ (𝐼 + 1) ∈ {(𝐼 + 1)}) |
| 13 | fvun1 7000 | . . 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 1463 | . 2 ⊢ (({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))‘(𝐼 + 1)) = ({〈(𝐼 + 1), 1〉}‘(𝐼 + 1)) |
| 15 | 3, 4 | fvsn 7201 | . 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 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 〈cop 4632 × cxp 5683 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ...cfz 13547 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: axlowdimlem14 28970 axlowdimlem16 28972 |
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