Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10736 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ℝcr 10530 ≤ cle 10670 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-pre-lttri 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 |
This theorem is referenced by: cjcn2 14950 abscvgcvg 15168 dvfsumlem3 24619 dvradcnv 25003 ppip1le 25732 dchrvmasumiflem2 26072 dchrisum0lem3 26089 rplogsum 26097 mudivsum 26100 dnibndlem6 33817 fltnltalem 39267 int-eqineqd 40536 sublevolico 42262 fourierdlem10 42395 fourierdlem12 42397 fourierdlem37 42422 fourierdlem48 42432 fourierdlem54 42438 fourierdlem79 42463 ioorrnopnxrlem 42584 hoidmvval0b 42865 hoidmv1lelem1 42866 hoidmvlelem2 42871 ovnhoi 42878 volico2 42916 ovolval5lem2 42928 vonioolem2 42956 lighneallem2 43764 fllog2 44621 |
Copyright terms: Public domain | W3C validator |