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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 12232. (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 3994 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℝ) |
5 | 4 | recnd 11286 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ) |
6 | supsubc.b | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | 6 | recnd 11286 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝐵 ∈ ℂ) |
9 | 5, 8 | negsubd 11623 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 + -𝐵) = (𝑣 − 𝐵)) |
10 | 9 | eqcomd 2740 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 − 𝐵) = (𝑣 + -𝐵)) |
11 | 10 | eqeq2d 2745 | . . . . . 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 2805 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
14 | eqidd 2735 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) | |
15 | 2, 13, 14 | 3eqtrd 2778 | . . 3 ⊢ (𝜑 → 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
16 | 15 | supeq1d 9483 | . 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 11687 | . . . 4 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
20 | eqid 2734 | . . . 4 ⊢ {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} | |
21 | 3, 17, 18, 19, 20 | supaddc 12232 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < )) |
22 | 21 | eqcomd 2740 | . 2 ⊢ (𝜑 → sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < ) = (sup(𝐴, ℝ, < ) + -𝐵)) |
23 | suprcl 12225 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
24 | 3, 17, 18, 23 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
25 | 24 | recnd 11286 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℂ) |
26 | 25, 7 | negsubd 11623 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = (sup(𝐴, ℝ, < ) − 𝐵)) |
27 | 16, 22, 26 | 3eqtrrd 2779 | 1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {cab 2711 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 (class class class)co 7430 supcsup 9477 ℂcc 11150 ℝcr 11151 + caddc 11155 < clt 11292 ≤ cle 11293 − cmin 11489 -cneg 11490 |
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 ax-pre-sup 11230 |
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-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 |
This theorem is referenced by: hoidmvlelem1 46550 |
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