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Mirrors > Home > ILE Home > Th. List > ltaddsub2 | GIF version |
Description: 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.) |
Ref | Expression |
---|---|
ltaddsub2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7979 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | recn 7979 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
3 | addcom 8129 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
4 | 1, 2, 3 | syl2an 289 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
5 | 4 | 3adant3 1019 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
6 | 5 | breq1d 4031 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ (𝐵 + 𝐴) < 𝐶)) |
7 | ltaddsub 8428 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) | |
8 | 7 | 3com12 1209 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) |
9 | 6, 8 | bitrd 188 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 (class class class)co 5900 ℂcc 7844 ℝcr 7845 + caddc 7849 < clt 8027 − cmin 8163 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-addass 7948 ax-distr 7950 ax-i2m1 7951 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 ax-pre-ltadd 7962 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-pnf 8029 df-mnf 8030 df-ltxr 8032 df-sub 8165 df-neg 8166 |
This theorem is referenced by: ltsub13 8435 ltaddsub2d 8538 addltmul 9190 absdiflt 11142 |
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