xleadd1d - Mathbox for Glauco Siliprandi

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Theorem xleadd1d 46480

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 13393	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_e 𝐶) ≤ (𝐵 +_e 𝐶))
6	1, 2, 3, 4, 5	syl31anc 1400	1 ⊢ (𝜑 → (𝐴 +_e 𝐶) ≤ (𝐵 +_e 𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2146 class class class wbr 5105 (class class class)co 7427 ℝ^*cxr 11352 ≤ cle 11354 +_e cxad 13249

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-id 5549 df-po 5562 df-so 5563 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7430 df-oprab 7431 df-mpo 7432 df-1st 8008 df-2nd 8009 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-xadd 13252

This theorem is used by: xle2addd 46487 infleinflem1 46520 sge0prle 47550

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