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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 12182. (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 3977 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℝ) |
5 | 4 | recnd 11243 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ) |
6 | supsubc.b | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | 6 | recnd 11243 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝐵 ∈ ℂ) |
9 | 5, 8 | negsubd 11578 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 + -𝐵) = (𝑣 − 𝐵)) |
10 | 9 | eqcomd 2732 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 − 𝐵) = (𝑣 + -𝐵)) |
11 | 10 | eqeq2d 2737 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑧 = (𝑣 − 𝐵) ↔ 𝑧 = (𝑣 + -𝐵))) |
12 | 11 | rexbidva 3170 | . . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵) ↔ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵))) |
13 | 12 | abbidv 2795 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
14 | eqidd 2727 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) | |
15 | 2, 13, 14 | 3eqtrd 2770 | . . 3 ⊢ (𝜑 → 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
16 | 15 | supeq1d 9440 | . 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 11642 | . . . 4 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
20 | eqid 2726 | . . . 4 ⊢ {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} | |
21 | 3, 17, 18, 19, 20 | supaddc 12182 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < )) |
22 | 21 | eqcomd 2732 | . 2 ⊢ (𝜑 → sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < ) = (sup(𝐴, ℝ, < ) + -𝐵)) |
23 | suprcl 12175 | . . . . 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 11243 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℂ) |
26 | 25, 7 | negsubd 11578 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = (sup(𝐴, ℝ, < ) − 𝐵)) |
27 | 16, 22, 26 | 3eqtrrd 2771 | 1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 (class class class)co 7404 supcsup 9434 ℂcc 11107 ℝcr 11108 + caddc 11112 < clt 11249 ≤ cle 11250 − cmin 11445 -cneg 11446 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 |
This theorem is referenced by: hoidmvlelem1 45865 |
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