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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 12127. (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 3945 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℝ) |
5 | 4 | recnd 11188 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ) |
6 | supsubc.b | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | 6 | recnd 11188 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | 7 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝐵 ∈ ℂ) |
9 | 5, 8 | negsubd 11523 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 + -𝐵) = (𝑣 − 𝐵)) |
10 | 9 | eqcomd 2739 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑣 − 𝐵) = (𝑣 + -𝐵)) |
11 | 10 | eqeq2d 2744 | . . . . . 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 2802 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
14 | eqidd 2734 | . . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) | |
15 | 2, 13, 14 | 3eqtrd 2777 | . . 3 ⊢ (𝜑 → 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}) |
16 | 15 | supeq1d 9387 | . 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 11587 | . . . 4 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
20 | eqid 2733 | . . . 4 ⊢ {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)} | |
21 | 3, 17, 18, 19, 20 | supaddc 12127 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < )) |
22 | 21 | eqcomd 2739 | . 2 ⊢ (𝜑 → sup({𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + -𝐵)}, ℝ, < ) = (sup(𝐴, ℝ, < ) + -𝐵)) |
23 | suprcl 12120 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
24 | 3, 17, 18, 23 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
25 | 24 | recnd 11188 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℂ) |
26 | 25, 7 | negsubd 11523 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + -𝐵) = (sup(𝐴, ℝ, < ) − 𝐵)) |
27 | 16, 22, 26 | 3eqtrrd 2778 | 1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3911 ∅c0 4283 class class class wbr 5106 (class class class)co 7358 supcsup 9381 ℂcc 11054 ℝcr 11055 + caddc 11059 < clt 11194 ≤ cle 11195 − cmin 11390 -cneg 11391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 |
This theorem is referenced by: hoidmvlelem1 44922 |
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