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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 3164 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℤ → (-𝑤 ∈ 𝐴 → -𝑤 ∈ ℤ)) | |
2 | recn 7974 | . . . . . . . . 9 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ) | |
3 | negid 8234 | . . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℂ → (𝑤 + -𝑤) = 0) | |
4 | 0z 9294 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℤ | |
5 | 3, 4 | eqeltrdi 2280 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ ℂ → (𝑤 + -𝑤) ∈ ℤ) |
6 | 5 | pm4.71i 391 | . . . . . . . . . 10 ⊢ (𝑤 ∈ ℂ ↔ (𝑤 ∈ ℂ ∧ (𝑤 + -𝑤) ∈ ℤ)) |
7 | zrevaddcl 9333 | . . . . . . . . . 10 ⊢ (-𝑤 ∈ ℤ → ((𝑤 ∈ ℂ ∧ (𝑤 + -𝑤) ∈ ℤ) ↔ 𝑤 ∈ ℤ)) | |
8 | 6, 7 | bitrid 192 | . . . . . . . . 9 ⊢ (-𝑤 ∈ ℤ → (𝑤 ∈ ℂ ↔ 𝑤 ∈ ℤ)) |
9 | 2, 8 | imbitrid 154 | . . . . . . . 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 254 | . . . . 5 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)) |
13 | simpr 110 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴) | |
14 | 13 | a1i 9 | . . . . 5 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)) |
15 | 12, 14 | jcad 307 | . . . 4 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) → (𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴))) |
16 | zre 9287 | . . . . 5 ⊢ (𝑤 ∈ ℤ → 𝑤 ∈ ℝ) | |
17 | 16 | anim1i 340 | . . . 4 ⊢ ((𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴)) |
18 | 15, 17 | impbid1 142 | . . 3 ⊢ (𝐴 ⊆ ℤ → ((𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴) ↔ (𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴))) |
19 | negeq 8180 | . . . . 5 ⊢ (𝑧 = 𝑤 → -𝑧 = -𝑤) | |
20 | 19 | eleq1d 2258 | . . . 4 ⊢ (𝑧 = 𝑤 → (-𝑧 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴)) |
21 | 20 | elrab 2908 | . . 3 ⊢ (𝑤 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ 𝐴)) |
22 | 20 | elrab 2908 | . . 3 ⊢ (𝑤 ∈ {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴} ↔ (𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴)) |
23 | 18, 21, 22 | 3bitr4g 223 | . 2 ⊢ (𝐴 ⊆ ℤ → (𝑤 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ 𝑤 ∈ {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴})) |
24 | 23 | eqrdv 2187 | 1 ⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 {crab 2472 ⊆ wss 3144 (class class class)co 5896 ℂcc 7839 ℝcr 7840 0cc0 7841 + caddc 7844 -cneg 8159 ℤcz 9283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-n0 9207 df-z 9284 |
This theorem is referenced by: (None) |
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