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Mirrors > Home > MPE Home > Th. List > shft2rab | Structured version Visualization version GIF version |
Description: If 𝐵 is a shift of 𝐴 by 𝐶, then 𝐴 is a shift of 𝐵 by -𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.) |
Ref | Expression |
---|---|
ovolshft.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ovolshft.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ovolshft.3 | ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) |
Ref | Expression |
---|---|
shft2rab | ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ (𝑦 − -𝐶) ∈ 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolshft.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | 1 | sseld 3567 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
3 | 2 | pm4.71rd 665 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
4 | recn 9905 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
5 | ovolshft.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | 5 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
7 | subneg 10209 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝑦 − -𝐶) = (𝑦 + 𝐶)) | |
8 | 4, 6, 7 | syl2anr 494 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − -𝐶) = (𝑦 + 𝐶)) |
9 | ovolshft.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) | |
10 | 9 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) |
11 | 8, 10 | eleq12d 2682 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 − -𝐶) ∈ 𝐵 ↔ (𝑦 + 𝐶) ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})) |
12 | id 22 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ) | |
13 | readdcl 9898 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑦 + 𝐶) ∈ ℝ) | |
14 | 12, 5, 13 | syl2anr 494 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝐶) ∈ ℝ) |
15 | oveq1 6556 | . . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 𝐶) → (𝑥 − 𝐶) = ((𝑦 + 𝐶) − 𝐶)) | |
16 | 15 | eleq1d 2672 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 + 𝐶) → ((𝑥 − 𝐶) ∈ 𝐴 ↔ ((𝑦 + 𝐶) − 𝐶) ∈ 𝐴)) |
17 | 16 | elrab3 3332 | . . . . . . 7 ⊢ ((𝑦 + 𝐶) ∈ ℝ → ((𝑦 + 𝐶) ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ↔ ((𝑦 + 𝐶) − 𝐶) ∈ 𝐴)) |
18 | 14, 17 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + 𝐶) ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ↔ ((𝑦 + 𝐶) − 𝐶) ∈ 𝐴)) |
19 | pncan 10166 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝑦 + 𝐶) − 𝐶) = 𝑦) | |
20 | 4, 6, 19 | syl2anr 494 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 + 𝐶) − 𝐶) = 𝑦) |
21 | 20 | eleq1d 2672 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((𝑦 + 𝐶) − 𝐶) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
22 | 11, 18, 21 | 3bitrd 293 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 − -𝐶) ∈ 𝐵 ↔ 𝑦 ∈ 𝐴)) |
23 | 22 | pm5.32da 671 | . . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℝ ∧ (𝑦 − -𝐶) ∈ 𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
24 | 3, 23 | bitr4d 270 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ (𝑦 − -𝐶) ∈ 𝐵))) |
25 | 24 | abbi2dv 2729 | . 2 ⊢ (𝜑 → 𝐴 = {𝑦 ∣ (𝑦 ∈ ℝ ∧ (𝑦 − -𝐶) ∈ 𝐵)}) |
26 | df-rab 2905 | . 2 ⊢ {𝑦 ∈ ℝ ∣ (𝑦 − -𝐶) ∈ 𝐵} = {𝑦 ∣ (𝑦 ∈ ℝ ∧ (𝑦 − -𝐶) ∈ 𝐵)} | |
27 | 25, 26 | syl6eqr 2662 | 1 ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ (𝑦 − -𝐶) ∈ 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 {crab 2900 ⊆ wss 3540 (class class class)co 6549 ℂcc 9813 ℝcr 9814 + caddc 9818 − cmin 10145 -cneg 10146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 |
This theorem is referenced by: ovolshft 23086 shftmbl 23113 |
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