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Theorem infmrp1 12738

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 12735	. 2 ⊢ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥
2		ltso 10721	. . . 4 ⊢ < Or ℝ
3	2	a1i 11	. . 3 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → < Or ℝ)
4		0red 10644	. . 3 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → 0 ∈ ℝ)
5		0red 10644	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 0 ∈ ℝ)
6		rpre 12398	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
7		rpge0 12403	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → 0 ≤ 𝑧)
8	5, 6, 7	lensymd 10791	. . . 4 ⊢ (𝑧 ∈ ℝ⁺ → ¬ 𝑧 < 0)
9	8	adantl 484	. . 3 ⊢ ((∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 ∧ 𝑧 ∈ ℝ⁺) → ¬ 𝑧 < 0)
10		elrp 12392	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧))
11		breq2 5070	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑧))
12	11	rexbidv 3297	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 ↔ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧))
13	12	rspcv 3618	. . . . 5 ⊢ (𝑧 ∈ ℝ⁺ → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧))
14	10, 13	sylbir 237	. . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 0 < 𝑧) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 → ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧))
15	14	impcom 410	. . 3 ⊢ ((∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑥 ∧ (𝑧 ∈ ℝ ∧ 0 < 𝑧)) → ∃𝑦 ∈ ℝ⁺ 𝑦 < 𝑧)
16	3, 4, 9, 15	eqinfd 8949	. 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 Or wor 5473 infcinf 8905 ℝcr 10536 0cc0 10537 < clt 10675 ℝ⁺crp 12390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-rp 12391

This theorem is referenced by: (None)

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