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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > supsubc | Structured version Visualization version GIF version |
Description: The supremum function distributes over subtraction in a sense similar to that in supaddc 12219. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
supsubc.a1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
supsubc.a2 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
supsubc.a3 | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
supsubc.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
supsubc.c | ⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)} |
Ref | Expression |
---|---|
supsubc | ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supsubc.c | . . . . 5 ⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)} | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)}) |
3 | supsubc.a1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
4 | 3 | sselda 3982 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℝ) |
5 | 4 | recnd 11280 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ) |
6 | supsubc.b | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | 6 | recnd 11280 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | 7 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝐵 ∈ ℂ) |
9 | 5, 8 | negsubd 11615 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 + -𝐵) = (𝑣 − 𝐵)) |
10 | 9 | eqcomd 2734 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 − 𝐵) = (𝑣 + -𝐵)) |
11 | 10 | eqeq2d 2739 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑧 = (𝑣 − 𝐵) ↔ 𝑧 = (𝑣 + -𝐵))) |
12 | 11 | rexbidva 3174 | . . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵) ↔ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵))) |
13 | 12 | abbidv 2797 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
14 | eqidd 2729 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) | |
15 | 2, 13, 14 | 3eqtrd 2772 | . . 3 ⊢ (𝜑 → 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
16 | 15 | supeq1d 9477 | . 2 ⊢ (𝜑 → sup(𝐶, ℝ, < ) = sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < )) |
17 | supsubc.a2 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
18 | supsubc.a3 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | |
19 | 6 | renegcld 11679 | . . . 4 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
20 | eqid 2728 | . . . 4 ⊢ {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} | |
21 | 3, 17, 18, 19, 20 | supaddc 12219 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < )) |
22 | 21 | eqcomd 2734 | . 2 ⊢ (𝜑 → sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < ) = (sup(𝐴, ℝ, < ) + -𝐵)) |
23 | suprcl 12212 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
24 | 3, 17, 18, 23 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
25 | 24 | recnd 11280 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℂ) |
26 | 25, 7 | negsubd 11615 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = (sup(𝐴, ℝ, < ) − 𝐵)) |
27 | 16, 22, 26 | 3eqtrrd 2773 | 1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2705 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 (class class class)co 7426 supcsup 9471 ℂcc 11144 ℝcr 11145 + caddc 11149 < clt 11286 ≤ cle 11287 − cmin 11482 -cneg 11483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 |
This theorem is referenced by: hoidmvlelem1 46012 |
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