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Mirrors > Home > MPE Home > Th. List > cnso | Structured version Visualization version GIF version |
Description: The complex numbers can be linearly ordered. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Ref | Expression |
---|---|
cnso | ⊢ ∃𝑥 𝑥 Or ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 11293 | . . . 4 ⊢ < Or ℝ | |
2 | eqid 2724 | . . . . . 6 ⊢ {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} | |
3 | f1oiso 7341 | . . . . . 6 ⊢ ((𝑎:ℝ–1-1-onto→ℂ ∧ {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)}) → 𝑎 Isom < , {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} (ℝ, ℂ)) | |
4 | 2, 3 | mpan2 688 | . . . . 5 ⊢ (𝑎:ℝ–1-1-onto→ℂ → 𝑎 Isom < , {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} (ℝ, ℂ)) |
5 | isoso 7338 | . . . . . 6 ⊢ (𝑎 Isom < , {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} (ℝ, ℂ) → ( < Or ℝ ↔ {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} Or ℂ)) | |
6 | soinxp 5748 | . . . . . 6 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} Or ℂ ↔ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) Or ℂ) | |
7 | 5, 6 | bitrdi 287 | . . . . 5 ⊢ (𝑎 Isom < , {⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} (ℝ, ℂ) → ( < Or ℝ ↔ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) Or ℂ)) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝑎:ℝ–1-1-onto→ℂ → ( < Or ℝ ↔ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) Or ℂ)) |
9 | 1, 8 | mpbii 232 | . . 3 ⊢ (𝑎:ℝ–1-1-onto→ℂ → ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) Or ℂ) |
10 | cnex 11188 | . . . . . 6 ⊢ ℂ ∈ V | |
11 | 10, 10 | xpex 7734 | . . . . 5 ⊢ (ℂ × ℂ) ∈ V |
12 | 11 | inex2 5309 | . . . 4 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) ∈ V |
13 | soeq1 5600 | . . . 4 ⊢ (𝑥 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) → (𝑥 Or ℂ ↔ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) Or ℂ)) | |
14 | 12, 13 | spcev 3588 | . . 3 ⊢ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑 ∈ ℝ ∃𝑒 ∈ ℝ ((𝑏 = (𝑎‘𝑑) ∧ 𝑐 = (𝑎‘𝑒)) ∧ 𝑑 < 𝑒)} ∩ (ℂ × ℂ)) Or ℂ → ∃𝑥 𝑥 Or ℂ) |
15 | 9, 14 | syl 17 | . 2 ⊢ (𝑎:ℝ–1-1-onto→ℂ → ∃𝑥 𝑥 Or ℂ) |
16 | rpnnen 16173 | . . . 4 ⊢ ℝ ≈ 𝒫 ℕ | |
17 | cpnnen 16175 | . . . 4 ⊢ ℂ ≈ 𝒫 ℕ | |
18 | 16, 17 | entr4i 9004 | . . 3 ⊢ ℝ ≈ ℂ |
19 | bren 8946 | . . 3 ⊢ (ℝ ≈ ℂ ↔ ∃𝑎 𝑎:ℝ–1-1-onto→ℂ) | |
20 | 18, 19 | mpbi 229 | . 2 ⊢ ∃𝑎 𝑎:ℝ–1-1-onto→ℂ |
21 | 15, 20 | exlimiiv 1926 | 1 ⊢ ∃𝑥 𝑥 Or ℂ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 ∃wrex 3062 ∩ cin 3940 𝒫 cpw 4595 class class class wbr 5139 {copab 5201 Or wor 5578 × cxp 5665 –1-1-onto→wf1o 6533 ‘cfv 6534 Isom wiso 6535 ≈ cen 8933 ℂcc 11105 ℝcr 11106 < clt 11247 ℕcn 12211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-acn 9934 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12976 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 |
This theorem is referenced by: aannenlem3 26207 |
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