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Mirrors > Home > MPE Home > Th. List > lemaxle | Structured version Visualization version GIF version |
Description: A real number which is less than or equal to a second real number is less than or equal to the maximum/supremum of the second real number and a third real number. (Contributed by AV, 8-Jun-2021.) |
Ref | Expression |
---|---|
lemaxle | ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | max2 13172 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) | |
2 | 1 | ancoms 458 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) |
3 | 2 | adantr 480 | . . 3 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) |
4 | simpr 484 | . . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ) | |
5 | simpll 764 | . . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ) | |
6 | ifcl 4568 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ) | |
7 | 6 | adantr 480 | . . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ) |
8 | letr 11312 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))) | |
9 | 4, 5, 7, 8 | syl3anc 1368 | . . 3 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))) |
10 | 3, 9 | mpan2d 691 | . 2 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))) |
11 | 10 | 3impia 1114 | 1 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ifcif 4523 class class class wbr 5141 ℝcr 11111 ≤ cle 11253 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 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-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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 |
This theorem is referenced by: (None) |
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