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Mirrors > Home > MPE Home > Th. List > sca2rab | Structured version Visualization version GIF version |
Description: If 𝐵 is a scale of 𝐴 by 𝐶, then 𝐴 is a scale of 𝐵 by 1 / 𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.) |
Ref | Expression |
---|---|
ovolsca.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ovolsca.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
ovolsca.3 | ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) |
Ref | Expression |
---|---|
sca2rab | ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolsca.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | 1 | sseld 3993 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
3 | 2 | pm4.71rd 562 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
4 | ovolsca.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) | |
5 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) |
6 | 5 | eleq2d 2824 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((1 / 𝐶) · 𝑦) ∈ 𝐵 ↔ ((1 / 𝐶) · 𝑦) ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})) |
7 | ovolsca.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ∈ ℝ+) |
9 | 8 | rprecred 13085 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (1 / 𝐶) ∈ ℝ) |
10 | remulcl 11237 | . . . . . . . 8 ⊢ (((1 / 𝐶) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((1 / 𝐶) · 𝑦) ∈ ℝ) | |
11 | 9, 10 | sylancom 588 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((1 / 𝐶) · 𝑦) ∈ ℝ) |
12 | oveq2 7438 | . . . . . . . . 9 ⊢ (𝑥 = ((1 / 𝐶) · 𝑦) → (𝐶 · 𝑥) = (𝐶 · ((1 / 𝐶) · 𝑦))) | |
13 | 12 | eleq1d 2823 | . . . . . . . 8 ⊢ (𝑥 = ((1 / 𝐶) · 𝑦) → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · ((1 / 𝐶) · 𝑦)) ∈ 𝐴)) |
14 | 13 | elrab3 3695 | . . . . . . 7 ⊢ (((1 / 𝐶) · 𝑦) ∈ ℝ → (((1 / 𝐶) · 𝑦) ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ↔ (𝐶 · ((1 / 𝐶) · 𝑦)) ∈ 𝐴)) |
15 | 11, 14 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((1 / 𝐶) · 𝑦) ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ↔ (𝐶 · ((1 / 𝐶) · 𝑦)) ∈ 𝐴)) |
16 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
17 | 16 | recnd 11286 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
18 | 8 | rpcnd 13076 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ∈ ℂ) |
19 | 8 | rpne0d 13079 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ≠ 0) |
20 | 17, 18, 19 | divrec2d 12044 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 / 𝐶) = ((1 / 𝐶) · 𝑦)) |
21 | 20 | oveq2d 7446 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · (𝑦 / 𝐶)) = (𝐶 · ((1 / 𝐶) · 𝑦))) |
22 | 17, 18, 19 | divcan2d 12042 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · (𝑦 / 𝐶)) = 𝑦) |
23 | 21, 22 | eqtr3d 2776 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · ((1 / 𝐶) · 𝑦)) = 𝑦) |
24 | 23 | eleq1d 2823 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝐶 · ((1 / 𝐶) · 𝑦)) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
25 | 6, 15, 24 | 3bitrd 305 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((1 / 𝐶) · 𝑦) ∈ 𝐵 ↔ 𝑦 ∈ 𝐴)) |
26 | 25 | pm5.32da 579 | . . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℝ ∧ ((1 / 𝐶) · 𝑦) ∈ 𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
27 | 3, 26 | bitr4d 282 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ ((1 / 𝐶) · 𝑦) ∈ 𝐵))) |
28 | 27 | eqabdv 2872 | . 2 ⊢ (𝜑 → 𝐴 = {𝑦 ∣ (𝑦 ∈ ℝ ∧ ((1 / 𝐶) · 𝑦) ∈ 𝐵)}) |
29 | df-rab 3433 | . 2 ⊢ {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵} = {𝑦 ∣ (𝑦 ∈ ℝ ∧ ((1 / 𝐶) · 𝑦) ∈ 𝐵)} | |
30 | 28, 29 | eqtr4di 2792 | 1 ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {cab 2711 {crab 3432 ⊆ wss 3962 (class class class)co 7430 ℝcr 11151 1c1 11153 · cmul 11157 / cdiv 11917 ℝ+crp 13031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-rp 13032 |
This theorem is referenced by: ovolsca 25563 |
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