![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eqreznegel | GIF version |
Description: Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
eqreznegel | ⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3096 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℤ → (-𝑤 ∈ 𝐴 → -𝑤 ∈ ℤ)) | |
2 | recn 7777 | . . . . . . . . 9 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ) | |
3 | negid 8033 | . . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℂ → (𝑤 + -𝑤) = 0) | |
4 | 0z 9089 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℤ | |
5 | 3, 4 | eqeltrdi 2231 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ ℂ → (𝑤 + -𝑤) ∈ ℤ) |
6 | 5 | pm4.71i 389 | . . . . . . . . . 10 ⊢ (𝑤 ∈ ℂ ↔ (𝑤 ∈ ℂ ∧ (𝑤 + -𝑤) ∈ ℤ)) |
7 | zrevaddcl 9128 | . . . . . . . . . 10 ⊢ (-𝑤 ∈ ℤ → ((𝑤 ∈ ℂ ∧ (𝑤 + -𝑤) ∈ ℤ) ↔ 𝑤 ∈ ℤ)) | |
8 | 6, 7 | syl5bb 191 | . . . . . . . . 9 ⊢ (-𝑤 ∈ ℤ → (𝑤 ∈ ℂ ↔ 𝑤 ∈ ℤ)) |
9 | 2, 8 | syl5ib 153 | . . . . . . . 8 ⊢ (-𝑤 ∈ ℤ → (𝑤 ∈ ℝ → 𝑤 ∈ ℤ)) |
10 | 1, 9 | syl6 33 | . . . . . . 7 ⊢ (𝐴 ⊆ ℤ → (-𝑤 ∈ 𝐴 → (𝑤 ∈ ℝ → 𝑤 ∈ ℤ))) |
11 | 10 | com23 78 | . . . . . 6 ⊢ (𝐴 ⊆ ℤ → (𝑤 ∈ ℝ → (-𝑤 ∈ 𝐴 → 𝑤 ∈ ℤ))) |
12 | 11 | impd 252 | . . . . 5 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)) |
13 | simpr 109 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴) | |
14 | 13 | a1i 9 | . . . . 5 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)) |
15 | 12, 14 | jcad 305 | . . . 4 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → (𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴))) |
16 | zre 9082 | . . . . 5 ⊢ (𝑤 ∈ ℤ → 𝑤 ∈ ℝ) | |
17 | 16 | anim1i 338 | . . . 4 ⊢ ((𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴)) |
18 | 15, 17 | impbid1 141 | . . 3 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) ↔ (𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴))) |
19 | negeq 7979 | . . . . 5 ⊢ (𝑧 = 𝑤 → -𝑧 = -𝑤) | |
20 | 19 | eleq1d 2209 | . . . 4 ⊢ (𝑧 = 𝑤 → (-𝑧 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴)) |
21 | 20 | elrab 2844 | . . 3 ⊢ (𝑤 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴)) |
22 | 20 | elrab 2844 | . . 3 ⊢ (𝑤 ∈ {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴} ↔ (𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴)) |
23 | 18, 21, 22 | 3bitr4g 222 | . 2 ⊢ (𝐴 ⊆ ℤ → (𝑤 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ 𝑤 ∈ {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴})) |
24 | 23 | eqrdv 2138 | 1 ⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 {crab 2421 ⊆ wss 3076 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 + caddc 7647 -cneg 7958 ℤcz 9078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |