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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 3982 | . . . . 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 2827 | . . . . . 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 13088 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (1 / 𝐶) ∈ ℝ) |
| 10 | remulcl 11240 | . . . . . . . 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 7439 | . . . . . . . . 9 ⊢ (𝑥 = ((1 / 𝐶) · 𝑦) → (𝐶 · 𝑥) = (𝐶 · ((1 / 𝐶) · 𝑦))) | |
| 13 | 12 | eleq1d 2826 | . . . . . . . 8 ⊢ (𝑥 = ((1 / 𝐶) · 𝑦) → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · ((1 / 𝐶) · 𝑦)) ∈ 𝐴)) |
| 14 | 13 | elrab3 3693 | . . . . . . 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 11289 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
| 18 | 8 | rpcnd 13079 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ∈ ℂ) |
| 19 | 8 | rpne0d 13082 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ≠ 0) |
| 20 | 17, 18, 19 | divrec2d 12047 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 / 𝐶) = ((1 / 𝐶) · 𝑦)) |
| 21 | 20 | oveq2d 7447 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · (𝑦 / 𝐶)) = (𝐶 · ((1 / 𝐶) · 𝑦))) |
| 22 | 17, 18, 19 | divcan2d 12045 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · (𝑦 / 𝐶)) = 𝑦) |
| 23 | 21, 22 | eqtr3d 2779 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · ((1 / 𝐶) · 𝑦)) = 𝑦) |
| 24 | 23 | eleq1d 2826 | . . . . . 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 2875 | . 2 ⊢ (𝜑 → 𝐴 = {𝑦 ∣ (𝑦 ∈ ℝ ∧ ((1 / 𝐶) · 𝑦) ∈ 𝐵)}) |
| 29 | df-rab 3437 | . 2 ⊢ {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵} = {𝑦 ∣ (𝑦 ∈ ℝ ∧ ((1 / 𝐶) · 𝑦) ∈ 𝐵)} | |
| 30 | 28, 29 | eqtr4di 2795 | 1 ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 {crab 3436 ⊆ wss 3951 (class class class)co 7431 ℝcr 11154 1c1 11156 · cmul 11160 / cdiv 11920 ℝ+crp 13034 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-rp 13035 |
| This theorem is referenced by: ovolsca 25550 |
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