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Mirrors > Home > ILE Home > Th. List > axltadd | GIF version |
Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 6799 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axltadd | ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A < B → (𝐶 + A) < (𝐶 + B))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-ltadd 6799 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A <ℝ B → (𝐶 + A) <ℝ (𝐶 + B))) | |
2 | ltxrlt 6882 | . . 3 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A < B ↔ A <ℝ B)) | |
3 | 2 | 3adant3 923 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A < B ↔ A <ℝ B)) |
4 | readdcl 6805 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ A ∈ ℝ) → (𝐶 + A) ∈ ℝ) | |
5 | readdcl 6805 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ B ∈ ℝ) → (𝐶 + B) ∈ ℝ) | |
6 | ltxrlt 6882 | . . . . 5 ⊢ (((𝐶 + A) ∈ ℝ ∧ (𝐶 + B) ∈ ℝ) → ((𝐶 + A) < (𝐶 + B) ↔ (𝐶 + A) <ℝ (𝐶 + B))) | |
7 | 4, 5, 6 | syl2an 273 | . . . 4 ⊢ (((𝐶 ∈ ℝ ∧ A ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ B ∈ ℝ)) → ((𝐶 + A) < (𝐶 + B) ↔ (𝐶 + A) <ℝ (𝐶 + B))) |
8 | 7 | 3impdi 1189 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ A ∈ ℝ ∧ B ∈ ℝ) → ((𝐶 + A) < (𝐶 + B) ↔ (𝐶 + A) <ℝ (𝐶 + B))) |
9 | 8 | 3coml 1110 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + A) < (𝐶 + B) ↔ (𝐶 + A) <ℝ (𝐶 + B))) |
10 | 1, 3, 9 | 3imtr4d 192 | 1 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A < B → (𝐶 + A) < (𝐶 + B))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 ∈ wcel 1390 class class class wbr 3755 (class class class)co 5455 ℝcr 6710 + caddc 6714 <ℝ cltrr 6715 < clt 6857 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-addrcl 6780 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-pnf 6859 df-mnf 6860 df-ltxr 6862 |
This theorem is referenced by: ltadd2 7212 nnge1 7718 |
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