infmrp1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > infmrp1		Structured version Visualization version GIF version

Theorem infmrp1 12553

Description: The infimum of the positive reals is 0. (Contributed by AV, 5-Sep-2020.)

**Assertion**
Ref	Expression
infmrp1	⊢ inf(ℝ⁺, ℝ, < ) = 0

Proof of Theorem infmrp1 Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpltrp 12550	. 2 ⊢ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥
2		ltso 10521	. . . 4 ⊢ < Or ℝ
3	2	a1i 11	. . 3 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → < Or ℝ)
4		0red 10443	. . 3 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → 0 ∈ ℝ)
5		0red 10443	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 0 ∈ ℝ)
6		rpre 12212	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
7		rpge0 12219	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 0 ≤ 𝑧)
8	5, 6, 7	lensymd 10591	. . . 4 ⊢ (𝑧 ∈ ℝ⁺ → ¬ 𝑧 < 0)
9	8	adantl 474	. . 3 ⊢ ((∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 ∧ 𝑧 ∈ ℝ⁺) → ¬ 𝑧 < 0)
10		elrp 12206	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧))
11		breq2 4933	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑧))
12	11	rexbidv 3242	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 ↔ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧))
13	12	rspcv 3531	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧))
14	10, 13	sylbir 227	. . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 0 < 𝑧) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧))
15	14	impcom 399	. . 3 ⊢ ((∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 ∧ (𝑧 ∈ ℝ ∧ 0 < 𝑧)) → ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧)
16	3, 4, 9, 15	eqinfd 8744	. 2 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → inf(ℝ⁺, ℝ, < ) = 0)
17	1, 16	ax-mp 5	1 ⊢ inf(ℝ⁺, ℝ, < ) = 0

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ∃wrex 3089 class class class wbr 4929 Or wor 5325 infcinf 8700 ℝcr 10334 0cc0 10335 < clt 10474 ℝ⁺crp 12204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-2 11503 df-rp 12205

This theorem is referenced by: (None)

W3C validator