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Mirrors > Home > MPE Home > Th. List > eqled | Structured version Visualization version GIF version |
Description: Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
eqled.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
eqled.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
eqled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqled.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | eqled.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | eqle 10899 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ℝcr 10693 ≤ cle 10833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-pre-lttri 10768 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 |
This theorem is referenced by: cjcn2 15126 abscvgcvg 15346 dvfsumlem3 24879 dvradcnv 25267 ppip1le 25997 dchrvmasumiflem2 26337 dchrisum0lem3 26354 rplogsum 26362 mudivsum 26365 dnibndlem6 34349 aks4d1p1p2 39760 fltnltalem 40143 int-eqineqd 41420 sublevolico 43143 fourierdlem10 43276 fourierdlem12 43278 fourierdlem37 43303 fourierdlem48 43313 fourierdlem54 43319 fourierdlem79 43344 ioorrnopnxrlem 43465 hoidmvval0b 43746 hoidmv1lelem1 43747 hoidmvlelem2 43752 ovnhoi 43759 volico2 43797 ovolval5lem2 43809 vonioolem2 43837 lighneallem2 44674 fllog2 45530 |
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