xleadd1d - Mathbox for Glauco Siliprandi

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Theorem xleadd1d 42541

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 12843	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))
6	1, 2, 3, 4, 5	syl31anc 1375	1 ⊢ (𝜑 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝ^*cxr 10866 ≤ cle 10868 +_𝑒 cxad 12702

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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-xadd 12705

This theorem is referenced by: xle2addd 42548 infleinflem1 42582 sge0prle 43614

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