Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > difico | Structured version Visualization version GIF version |
Description: The difference between two closed-below, open-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
Ref | Expression |
---|---|
difico | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icodisj 12850 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅) | |
2 | undif4 4412 | . . . 4 ⊢ (((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶))) |
4 | 3 | adantr 481 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶))) |
5 | difid 4327 | . . . . 5 ⊢ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶)) = ∅ | |
6 | 5 | uneq2i 4133 | . . . 4 ⊢ ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = ((𝐴[,)𝐵) ∪ ∅) |
7 | un0 4341 | . . . 4 ⊢ ((𝐴[,)𝐵) ∪ ∅) = (𝐴[,)𝐵) | |
8 | 6, 7 | eqtri 2841 | . . 3 ⊢ ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (𝐴[,)𝐵) |
9 | 8 | a1i 11 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (𝐴[,)𝐵)) |
10 | icoun 12849 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴[,)𝐶)) | |
11 | 10 | difeq1d 4095 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶)) = ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶))) |
12 | 4, 9, 11 | 3eqtr3rd 2862 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 class class class wbr 5057 (class class class)co 7145 ℝ*cxr 10662 ≤ cle 10664 [,)cico 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ico 12732 |
This theorem is referenced by: sxbrsigalem2 31443 |
Copyright terms: Public domain | W3C validator |