xrletr - Metamath Proof Explorer

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Theorem xrletr 13179

Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

**Assertion**
Ref	Expression
xrletr	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶))

**Proof of Theorem xrletr**
Step	Hyp	Ref	Expression
1		xrleloe 13165	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
2	1	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
3	2	adantr 480	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
4		xrlelttr 13177	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
5		xrltle 13170	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
6	5	3adant2 1131	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
7	4, 6	syld 47	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 ≤ 𝐶))
8	7	expdimp 452	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐵 < 𝐶 → 𝐴 ≤ 𝐶))
9		breq2 5128	. . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ 𝐶))
10	9	biimpcd 249	. . . . 5 ⊢ (𝐴 ≤ 𝐵 → (𝐵 = 𝐶 → 𝐴 ≤ 𝐶))
11	10	adantl 481	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐵 = 𝐶 → 𝐴 ≤ 𝐶))
12	8, 11	jaod 859	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → ((𝐵 < 𝐶 ∨ 𝐵 = 𝐶) → 𝐴 ≤ 𝐶))
13	3, 12	sylbid 240	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐵 ≤ 𝐶 → 𝐴 ≤ 𝐶))
14	13	expimpd 453	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5124 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-pre-lttri 11208 ax-pre-lttrn 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280

This theorem is referenced by: xrletrd 13183 xrmaxle 13204 xrlemin 13205 xralrple 13226 xle2add 13280 icc0 13415 iccss 13436 icossico 13438 iccss2 13439 iccssico 13440 icoun 13497 snunico 13501 snunioc 13502 limsupgord 15493 limsupgre 15502 limsupbnd1 15503 limsupbnd2 15504 ramtlecl 17025 letsr 18608 leordtval2 23155 lecldbas 23162 imasdsf1olem 24317 stdbdxmet 24459 ovolmge0 25435 itg2le 25697 itg2seq 25700 plypf1 26174 pntleml 27579 ewlkle 29590 upgrewlkle2 29591 nmopge0 31897 nmfnge0 31913 xrstos 33007 xrge0omnd 33084 elicc3 36340 tan2h 37641 mblfinlem3 37688 mblfinlem4 37689 itg2addnclem 37700 monoordxrv 45475 liminfgord 45750

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