xle2add - Intuitionistic Logic Explorer

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Theorem xle2add 10021

Description: Extended real version of le2add 8537. (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 10015	. . . 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 1250	. 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 10016	. . . 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 1250	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
11		xaddcl 10002	. . . 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 10002	. . . 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 10002	. . . 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 9950	. . 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 1250	. 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 981 ∈ wcel 2177 class class class wbr 4051 (class class class)co 5957 ℝ^*cxr 8126 ≤ cle 8128 +_𝑒 cxad 9912

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-addcom 8045 ax-addass 8047 ax-i2m1 8050 ax-0id 8053 ax-rnegex 8054 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-id 4348 df-po 4351 df-iso 4352 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-fv 5288 df-ov 5960 df-oprab 5961 df-mpo 5962 df-1st 6239 df-2nd 6240 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-xadd 9915

This theorem is referenced by: xrbdtri 11662 xmetxp 15054

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