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Mirrors > Home > MPE Home > Th. List > ifle | Structured version Visualization version GIF version |
Description: An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.) |
Ref | Expression |
---|---|
ifle | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) → if(𝜑, 𝐴, 𝐵) ≤ if(𝜓, 𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll1 1210 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ 𝜑) → 𝐴 ∈ ℝ) | |
2 | 1 | leidd 11804 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ 𝜑) → 𝐴 ≤ 𝐴) |
3 | iftrue 4530 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴) |
5 | id 22 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
6 | 5 | imp 406 | . . . . 5 ⊢ (((𝜑 → 𝜓) ∧ 𝜑) → 𝜓) |
7 | 6 | adantll 713 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ 𝜑) → 𝜓) |
8 | 7 | iftrued 4532 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ 𝜑) → if(𝜓, 𝐴, 𝐵) = 𝐴) |
9 | 2, 4, 8 | 3brtr4d 5174 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) ≤ if(𝜓, 𝐴, 𝐵)) |
10 | iffalse 4533 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
11 | 10 | adantl 481 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐵) |
12 | simpll3 1212 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ ¬ 𝜑) → 𝐵 ≤ 𝐴) | |
13 | simpll2 1211 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ ¬ 𝜑) → 𝐵 ∈ ℝ) | |
14 | 13 | leidd 11804 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ ¬ 𝜑) → 𝐵 ≤ 𝐵) |
15 | breq2 5146 | . . . . 5 ⊢ (𝐴 = if(𝜓, 𝐴, 𝐵) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ if(𝜓, 𝐴, 𝐵))) | |
16 | breq2 5146 | . . . . 5 ⊢ (𝐵 = if(𝜓, 𝐴, 𝐵) → (𝐵 ≤ 𝐵 ↔ 𝐵 ≤ if(𝜓, 𝐴, 𝐵))) | |
17 | 15, 16 | ifboth 4563 | . . . 4 ⊢ ((𝐵 ≤ 𝐴 ∧ 𝐵 ≤ 𝐵) → 𝐵 ≤ if(𝜓, 𝐴, 𝐵)) |
18 | 12, 14, 17 | syl2anc 583 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ ¬ 𝜑) → 𝐵 ≤ if(𝜓, 𝐴, 𝐵)) |
19 | 11, 18 | eqbrtrd 5164 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) ≤ if(𝜓, 𝐴, 𝐵)) |
20 | 9, 19 | pm2.61dan 812 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) → if(𝜑, 𝐴, 𝐵) ≤ if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ifcif 4524 class class class wbr 5142 ℝcr 11131 ≤ cle 11273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-pre-lttri 11206 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 |
This theorem is referenced by: rpnnen2lem4 16187 itg2cnlem2 25685 |
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