xleadd2d - Mathbox for Glauco Siliprandi

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Theorem xleadd2d 41615

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

**Hypotheses**
Ref	Expression
xleadd2d.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xleadd2d.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
xleadd2d.3	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
xleadd2d.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
xleadd2d	⊢ (𝜑 → (𝐶 +_𝑒 𝐴) ≤ (𝐶 +_𝑒 𝐵))

**Proof of Theorem xleadd2d**
Step	Hyp	Ref	Expression
1		xleadd2d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xleadd2d.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		xleadd2d.3	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
4		xleadd2d.4	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		xleadd2a 12648	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐶 +_𝑒 𝐴) ≤ (𝐶 +_𝑒 𝐵))
6	1, 2, 3, 4, 5	syl31anc 1369	1 ⊢ (𝜑 → (𝐶 +_𝑒 𝐴) ≤ (𝐶 +_𝑒 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝ^*cxr 10674 ≤ cle 10676 +_𝑒 cxad 12506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-xadd 12509

This theorem is referenced by: xle2addd 41624 meaunle 42766

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