xrletr - Metamath Proof Explorer

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Theorem xrletr 11865

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 11853	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
2	1	3adant1 1072	. . . 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 11863	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
5		xrltle 11858	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
6	5	3adant2 1073	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
7	4, 6	syld 46	. . . . 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 4587	. . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ 𝐶))
10	9	biimpcd 238	. . . . 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 394	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → ((𝐵 < 𝐶 ∨ 𝐵 = 𝐶) → 𝐴 ≤ 𝐶))
13	3, 12	sylbid 229	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐵 ≤ 𝐶 → 𝐴 ≤ 𝐶))
14	13	expimpd 627	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954

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-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-po 4959 df-so 4960 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-xr 9957 df-ltxr 9958 df-le 9959

This theorem is referenced by: xrletrd 11869 xrmaxle 11888 xrlemin 11889 xralrple 11910 xle2add 11961 icc0 12094 iccss 12112 icossico 12114 iccss2 12115 iccssico 12116 icoun 12167 snunico 12170 snunioc 12171 limsupgord 14051 limsupgre 14060 limsupbnd1 14061 limsupbnd2 14062 ramtlecl 15542 letsr 17050 leordtval2 20826 lecldbas 20833 imasdsf1olem 21988 stdbdxmet 22130 ovolmge0 23052 itg2le 23312 itg2seq 23315 plypf1 23772 pntleml 25100 nmopge0 28154 nmfnge0 28170 xrstos 29010 xrge0omnd 29042 elicc3 31481 tan2h 32571 mblfinlem3 32618 mblfinlem4 32619 itg2addnclem 32631 ewlkle 40807 upgrewlkle2 40808

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