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Mirrors > Home > MPE Home > Th. List > axlowdimlem4 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26040. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.) |
Ref | Expression |
---|---|
axlowdimlem4.1 | ⊢ 𝐴 ∈ ℝ |
axlowdimlem4.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
axlowdimlem4 | ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ne2 11432 | . . . 4 ⊢ 1 ≠ 2 | |
2 | 1ex 10227 | . . . . 5 ⊢ 1 ∈ V | |
3 | 2ex 11284 | . . . . 5 ⊢ 2 ∈ V | |
4 | axlowdimlem4.1 | . . . . . 6 ⊢ 𝐴 ∈ ℝ | |
5 | 4 | elexi 3353 | . . . . 5 ⊢ 𝐴 ∈ V |
6 | axlowdimlem4.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
7 | 6 | elexi 3353 | . . . . 5 ⊢ 𝐵 ∈ V |
8 | 2, 3, 5, 7 | fpr 6584 | . . . 4 ⊢ (1 ≠ 2 → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶{𝐴, 𝐵}) |
9 | 1, 8 | ax-mp 5 | . . 3 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶{𝐴, 𝐵} |
10 | 1z 11599 | . . . . . 6 ⊢ 1 ∈ ℤ | |
11 | fzpr 12589 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
13 | df-2 11271 | . . . . . 6 ⊢ 2 = (1 + 1) | |
14 | 13 | oveq2i 6824 | . . . . 5 ⊢ (1...2) = (1...(1 + 1)) |
15 | 13 | preq2i 4416 | . . . . 5 ⊢ {1, 2} = {1, (1 + 1)} |
16 | 12, 14, 15 | 3eqtr4i 2792 | . . . 4 ⊢ (1...2) = {1, 2} |
17 | 16 | feq2i 6198 | . . 3 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶{𝐴, 𝐵} ↔ {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶{𝐴, 𝐵}) |
18 | 9, 17 | mpbir 221 | . 2 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶{𝐴, 𝐵} |
19 | 4, 6 | pm3.2i 470 | . . 3 ⊢ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
20 | 5, 7 | prss 4496 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ↔ {𝐴, 𝐵} ⊆ ℝ) |
21 | 19, 20 | mpbi 220 | . 2 ⊢ {𝐴, 𝐵} ⊆ ℝ |
22 | fss 6217 | . 2 ⊢ (({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶{𝐴, 𝐵} ∧ {𝐴, 𝐵} ⊆ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ) | |
23 | 18, 21, 22 | mp2an 710 | 1 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ⊆ wss 3715 {cpr 4323 〈cop 4327 ⟶wf 6045 (class class class)co 6813 ℝcr 10127 1c1 10129 + caddc 10131 2c2 11262 ℤcz 11569 ...cfz 12519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 |
This theorem is referenced by: axlowdimlem5 26025 axlowdimlem6 26026 axlowdimlem17 26037 |
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