xleadd1d - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xleadd1d		Structured version Visualization version GIF version

Theorem xleadd1d 41595

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 12645	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))
6	1, 2, 3, 4, 5	syl31anc 1369	1 ⊢ (𝜑 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5065 (class class class)co 7155 ℝ^*cxr 10673 ≤ cle 10675 +_𝑒 cxad 12504

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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-xadd 12507

This theorem is referenced by: xle2addd 41602 infleinflem1 41636 sge0prle 42682

W3C validator