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Mirrors > Home > MPE Home > Th. List > axltadd | Structured version Visualization version GIF version |
Description: Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 9891 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 9891 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) | |
2 | ltxrlt 9987 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) | |
3 | 2 | 3adant3 1074 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) |
4 | readdcl 9898 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 + 𝐴) ∈ ℝ) | |
5 | readdcl 9898 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ) | |
6 | ltxrlt 9987 | . . . . 5 ⊢ (((𝐶 + 𝐴) ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → ((𝐶 + 𝐴) < (𝐶 + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) | |
7 | 4, 5, 6 | syl2an 493 | . . . 4 ⊢ (((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → ((𝐶 + 𝐴) < (𝐶 + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) |
8 | 7 | 3impdi 1373 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 + 𝐴) < (𝐶 + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) |
9 | 8 | 3coml 1264 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) < (𝐶 + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) |
10 | 1, 3, 9 | 3imtr4d 282 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 + caddc 9818 <ℝ cltrr 9819 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-addrcl 9876 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: ltadd2 10020 nnge1 10923 ltoddhalfle 14923 sqrtpwpw2p 39988 |
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