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Mirrors > Home > MPE Home > Th. List > axlowdimlem4 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26434. 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 11699 | . . . 4 ⊢ 1 ≠ 2 | |
2 | 1ex 10490 | . . . . 5 ⊢ 1 ∈ V | |
3 | 2ex 11568 | . . . . 5 ⊢ 2 ∈ V | |
4 | axlowdimlem4.1 | . . . . . 6 ⊢ 𝐴 ∈ ℝ | |
5 | 4 | elexi 3459 | . . . . 5 ⊢ 𝐴 ∈ V |
6 | axlowdimlem4.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
7 | 6 | elexi 3459 | . . . . 5 ⊢ 𝐵 ∈ V |
8 | 2, 3, 5, 7 | fpr 6786 | . . . 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 | fz12pr 12818 | . . . 4 ⊢ (1...2) = {1, 2} | |
11 | 10 | feq2i 6381 | . . 3 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶{𝐴, 𝐵} ↔ {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶{𝐴, 𝐵}) |
12 | 9, 11 | mpbir 232 | . 2 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶{𝐴, 𝐵} |
13 | 4, 6 | pm3.2i 471 | . . 3 ⊢ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
14 | 5, 7 | prss 4666 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ↔ {𝐴, 𝐵} ⊆ ℝ) |
15 | 13, 14 | mpbi 231 | . 2 ⊢ {𝐴, 𝐵} ⊆ ℝ |
16 | fss 6402 | . 2 ⊢ (({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶{𝐴, 𝐵} ∧ {𝐴, 𝐵} ⊆ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ) | |
17 | 12, 15, 16 | mp2an 688 | 1 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2083 ≠ wne 2986 ⊆ wss 3865 {cpr 4480 〈cop 4484 ⟶wf 6228 (class class class)co 7023 ℝcr 10389 1c1 10391 2c2 11546 ...cfz 12746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 |
This theorem is referenced by: axlowdimlem5 26419 axlowdimlem6 26420 axlowdimlem17 26431 |
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