xle2add - Intuitionistic Logic Explorer

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Theorem xle2add 9954

Description: Extended real version of le2add 8471. (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 527	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐴 ∈ ℝ^)
2		simprl 529	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐶 ∈ ℝ^)
3		simplr 528	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐵 ∈ ℝ^)
4		xleadd1a 9948	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐶) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵))
5	4	ex 115	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
6	1, 2, 3, 5	syl3anc 1249	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
7		simprr 531	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐷 ∈ ℝ^)
8		xleadd2a 9949	. . . 4 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐷) → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷))
9	8	ex 115	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
10	3, 7, 2, 9	syl3anc 1249	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
11		xaddcl 9935	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	11	adantr 276	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
13		xaddcl 9935	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 +_𝑒 𝐵) ∈ ℝ^)
14	2, 3, 13	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐶 +_𝑒 𝐵) ∈ ℝ^)
15		xaddcl 9935	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
16	15	adantl 277	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
17		xrletr 9883	. . 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 1249	. 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 295	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 ∈ wcel 2167 class class class wbr 4033 (class class class)co 5922 ℝ^*cxr 8060 ≤ cle 8062 +_𝑒 cxad 9845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-addcom 7979 ax-addass 7981 ax-i2m1 7984 ax-0id 7987 ax-rnegex 7988 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-po 4331 df-iso 4332 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-xadd 9848

This theorem is referenced by: xrbdtri 11441 xmetxp 14743

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