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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 11363 | . 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 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: cjcn2 15636 abscvgcvg 15855 dvfsumlem3 26069 dvradcnv 26464 ppip1le 27204 dchrvmasumiflem2 27546 dchrisum0lem3 27563 rplogsum 27571 mudivsum 27574 dnibndlem6 36484 aks4d1p1p2 42071 unitscyglem4 42199 fltnltalem 42672 int-eqineqd 44203 sublevolico 45999 fourierdlem10 46132 fourierdlem12 46134 fourierdlem37 46159 fourierdlem48 46169 fourierdlem54 46175 fourierdlem79 46200 ioorrnopnxrlem 46321 hoidmvval0b 46605 hoidmv1lelem1 46606 hoidmvlelem2 46611 ovnhoi 46618 volico2 46656 ovolval5lem2 46668 vonioolem2 46696 lighneallem2 47593 fllog2 48489 |
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