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Mirrors > Home > MPE Home > Th. List > xrltletrd | Structured version Visualization version GIF version |
Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xrlttrd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrlttrd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrlttrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
xrltletrd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
xrltletrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
xrltletrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltletrd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | xrltletrd.5 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
3 | xrlttrd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrlttrd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrlttrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
6 | xrltletr 12553 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1367 | . 2 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 697 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 class class class wbr 5069 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-pre-lttri 10614 ax-pre-lttrn 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 |
This theorem is referenced by: xlt2add 12656 xadddi2 12693 supxrre 12723 infxrre 12732 ixxlb 12763 elicore 12792 elico2 12803 elicc2 12804 caucvgrlem 15032 isnzr2hash 20040 xrsdsreclblem 20594 xblss2ps 23014 xblss2 23015 tgioo 23407 xrge0tsms 23445 xrhmeo 23553 ovoliunlem1 24106 ovoliun 24109 ioombl1lem2 24163 vitalilem4 24215 itg2monolem2 24355 itg2gt0 24364 dvferm1lem 24584 dvferm2lem 24586 lhop1lem 24613 pserdvlem2 25019 abelthlem3 25024 logtayl 25246 xrge0tsmsd 30696 esum2d 31356 usgrcyclgt2v 32382 relowlssretop 34648 itg2gt0cn 34951 areacirclem5 34990 xrge0nemnfd 41606 supxrgere 41607 supxrgelem 41611 infrpge 41625 xrralrecnnge 41668 supxrunb3 41678 icoopn 41807 limsupre 41928 limsupre3lem 42019 xlimpnfv 42125 fourierdlem27 42426 fourierdlem87 42485 gsumge0cl 42660 sge0pr 42683 sge0ssre 42686 sge0xaddlem1 42722 meaiuninc3v 42773 pimiooltgt 42996 pimdecfgtioc 43000 preimageiingt 43005 |
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