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Mirrors > Home > ILE Home > Th. List > ltletrd | GIF version |
Description: Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.) |
Ref | Expression |
---|---|
ltadd2d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltadd2d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd2d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltletrd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
ltletrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
ltletrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | ltletrd.5 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
3 | ltadd2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ltadd2d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | ltadd2d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | ltletr 8049 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1238 | . 2 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 433 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4005 ℝcr 7812 < clt 7994 ≤ cle 7995 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-pre-ltwlin 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 |
This theorem is referenced by: lelttrdi 8385 lediv12a 8853 btwnapz 9385 rpgecl 9684 fznatpl1 10078 elfz1b 10092 exbtwnzlemstep 10250 ceiqle 10315 modqabs 10359 mulp1mod1 10367 seq3f1olemqsumk 10501 expgt1 10560 leexp2a 10575 bernneq3 10645 expnbnd 10646 nn0opthlem2d 10703 cvg1nlemres 10996 resqrexlemlo 11024 resqrexlemnmsq 11028 resqrexlemga 11034 abssubap0 11101 icodiamlt 11191 rpmaxcl 11234 reccn2ap 11323 divcnv 11507 cvgratnnlembern 11533 cvgratnnlemabsle 11537 fprodntrivap 11594 efcllemp 11668 sin01bnd 11767 cos01bnd 11768 sin01gt0 11771 cos12dec 11777 eirraplem 11786 dvdslelemd 11851 dvdsbnd 11959 isprm5 12144 1arith 12367 znnen 12401 nninfdclemp1 12453 cnopnap 14179 dedekindeulemlu 14184 suplociccreex 14187 dedekindicclemlu 14193 dedekindicc 14196 ivthinclemlopn 14199 limcimolemlt 14218 limccnp2lem 14230 coseq00topi 14341 cosordlem 14355 logdivlti 14387 |
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