xleadd1d - Mathbox for Glauco Siliprandi

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Theorem xleadd1d 42758

Description: Addition of extended reals preserves the "less than or equal to" relation, in the left slot. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
xleadd1d.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xleadd1d.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
xleadd1d.3	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
xleadd1d.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
xleadd1d	⊢ (𝜑 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))

**Proof of Theorem xleadd1d**
Step	Hyp	Ref	Expression
1		xleadd1d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xleadd1d.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		xleadd1d.3	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
4		xleadd1d.4	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		xleadd1a 12916	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))
6	1, 2, 3, 4, 5	syl31anc 1371	1 ⊢ (𝜑 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5070 (class class class)co 7255 ℝ^*cxr 10939 ≤ cle 10941 +_𝑒 cxad 12775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-xadd 12778

This theorem is referenced by: xle2addd 42765 infleinflem1 42799 sge0prle 43829

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