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Theorem xrltmin 13339

Description: Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.)

**Assertion**
Ref	Expression
xrltmin	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))

**Proof of Theorem xrltmin**
Step	Hyp	Ref	Expression
1		xrmin1 13334	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
2	1	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
3		simp1 1154	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		ifcl 4531	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
5	4	3adant1 1148	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
6		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
7		xrltletr 13313	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴 < 𝐵))
8	3, 5, 6, 7	syl3anc 1399	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴 < 𝐵))
9	2, 8	mpan2d 707	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐵))
10		xrmin2 13335	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
11	10	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
12		xrltletr 13313	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴 < 𝐶))
13	5, 12	syld3an2 1439	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴 < 𝐶))
14	11, 13	mpan2d 707	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐶))
15	9, 14	jcad 522	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
16		breq2 5110	. . 3 ⊢ (𝐵 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
17		breq2 5110	. . 3 ⊢ (𝐶 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐶 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
18	16, 17	ifboth 4525	. 2 ⊢ ((𝐴 < 𝐵 ∧ 𝐴 < 𝐶) → 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶))
19	15, 18	impbid1 228	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2147 ifcif 4485 class class class wbr 5106 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-pre-lttri 11301 ax-pre-lttrn 11302

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376

This theorem is used by: ltmin 13351 iooin 13537 blin 24770 lhop1 26364 ioondisj1 46711

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