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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 11361 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ℝcr 11152 ≤ cle 11294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-pre-lttri 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 |
This theorem is referenced by: cjcn2 15633 abscvgcvg 15852 dvfsumlem3 26084 dvradcnv 26479 ppip1le 27219 dchrvmasumiflem2 27561 dchrisum0lem3 27578 rplogsum 27586 mudivsum 27589 dnibndlem6 36466 aks4d1p1p2 42052 unitscyglem4 42180 fltnltalem 42649 int-eqineqd 44180 sublevolico 45940 fourierdlem10 46073 fourierdlem12 46075 fourierdlem37 46100 fourierdlem48 46110 fourierdlem54 46116 fourierdlem79 46141 ioorrnopnxrlem 46262 hoidmvval0b 46546 hoidmv1lelem1 46547 hoidmvlelem2 46552 ovnhoi 46559 volico2 46597 ovolval5lem2 46609 vonioolem2 46637 lighneallem2 47531 fllog2 48418 |
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