xle2add - Metamath Proof Explorer

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Theorem xle2add 12993

Description: Extended real version of le2add 11457. (Contributed by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
xle2add	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))

**Proof of Theorem xle2add**
Step	Hyp	Ref	Expression
1		simpll 764	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐴 ∈ ℝ^)
2		simprl 768	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐶 ∈ ℝ^)
3		simplr 766	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐵 ∈ ℝ^)
4		xleadd1a 12987	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐶) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵))
5	4	ex 413	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
6	1, 2, 3, 5	syl3anc 1370	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
7		simprr 770	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐷 ∈ ℝ^)
8		xleadd2a 12988	. . . 4 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐷) → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷))
9	8	ex 413	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
10	3, 7, 2, 9	syl3anc 1370	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
11		xaddcl 12973	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	11	adantr 481	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
13		xaddcl 12973	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 +_𝑒 𝐵) ∈ ℝ^)
14	2, 3, 13	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐶 +_𝑒 𝐵) ∈ ℝ^)
15		xaddcl 12973	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
16	15	adantl 482	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
17		xrletr 12892	. . 3 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ (𝐶 +_𝑒 𝐵) ∈ ℝ^* ∧ (𝐶 +_𝑒 𝐷) ∈ ℝ^*) → (((𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵) ∧ (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
18	12, 14, 16, 17	syl3anc 1370	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (((𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵) ∧ (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
19	6, 10, 18	syl2and 608	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝ^*cxr 11008 ≤ cle 11010 +_𝑒 cxad 12846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-xadd 12849

This theorem is referenced by: metnrmlem3 24024 xraddge02 31079 xrofsup 31090 esumpmono 32047 xadd0ge 42859 sge0split 43947

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