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Mirrors > Home > MPE Home > Th. List > creui | Structured version Visualization version GIF version |
Description: The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
creui | ⊢ (𝐴 ∈ ℂ → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 10641 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑧 ∈ ℝ ∃𝑤 ∈ ℝ 𝐴 = (𝑧 + (i · 𝑤))) | |
2 | simpr 487 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ) | |
3 | eqcom 2831 | . . . . . . . . . 10 ⊢ ((𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)) ↔ (𝑥 + (i · 𝑦)) = (𝑧 + (i · 𝑤))) | |
4 | cru 11633 | . . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ)) → ((𝑥 + (i · 𝑦)) = (𝑧 + (i · 𝑤)) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | |
5 | 4 | ancoms 461 | . . . . . . . . . 10 ⊢ (((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 + (i · 𝑦)) = (𝑧 + (i · 𝑤)) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) |
6 | 3, 5 | syl5bb 285 | . . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) |
7 | 6 | anass1rs 653 | . . . . . . . 8 ⊢ ((((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → ((𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) |
8 | 7 | rexbidva 3299 | . . . . . . 7 ⊢ (((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) ∧ 𝑦 ∈ ℝ) → (∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) |
9 | biidd 264 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑤 ↔ 𝑦 = 𝑤)) | |
10 | 9 | ceqsrexv 3652 | . . . . . . . 8 ⊢ (𝑧 ∈ ℝ → (∃𝑥 ∈ ℝ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ 𝑦 = 𝑤)) |
11 | 10 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) ∧ 𝑦 ∈ ℝ) → (∃𝑥 ∈ ℝ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ 𝑦 = 𝑤)) |
12 | 8, 11 | bitrd 281 | . . . . . 6 ⊢ (((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) ∧ 𝑦 ∈ ℝ) → (∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)) ↔ 𝑦 = 𝑤)) |
13 | 12 | ralrimiva 3185 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ∀𝑦 ∈ ℝ (∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)) ↔ 𝑦 = 𝑤)) |
14 | reu6i 3722 | . . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ ∀𝑦 ∈ ℝ (∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)) ↔ 𝑦 = 𝑤)) → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦))) | |
15 | 2, 13, 14 | syl2anc 586 | . . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦))) |
16 | eqeq1 2828 | . . . . . 6 ⊢ (𝐴 = (𝑧 + (i · 𝑤)) → (𝐴 = (𝑥 + (i · 𝑦)) ↔ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)))) | |
17 | 16 | rexbidv 3300 | . . . . 5 ⊢ (𝐴 = (𝑧 + (i · 𝑤)) → (∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)))) |
18 | 17 | reubidv 3392 | . . . 4 ⊢ (𝐴 = (𝑧 + (i · 𝑤)) → (∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)) ↔ ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ (𝑧 + (i · 𝑤)) = (𝑥 + (i · 𝑦)))) |
19 | 15, 18 | syl5ibrcom 249 | . . 3 ⊢ ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝐴 = (𝑧 + (i · 𝑤)) → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))) |
20 | 19 | rexlimivv 3295 | . 2 ⊢ (∃𝑧 ∈ ℝ ∃𝑤 ∈ ℝ 𝐴 = (𝑧 + (i · 𝑤)) → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) |
21 | 1, 20 | syl 17 | 1 ⊢ (𝐴 ∈ ℂ → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 ∃!wreu 3143 (class class class)co 7159 ℂcc 10538 ℝcr 10539 ici 10542 + caddc 10543 · cmul 10545 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 |
This theorem is referenced by: (None) |
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