sqrlem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sqrlem4		Structured version Visualization version GIF version

Theorem sqrlem4 14605

Description: Lemma for 01sqrex 14609. (Contributed by Mario Carneiro, 10-Jul-2013.)

**Hypotheses**
Ref	Expression
sqrlem1.1	⊢ 𝑆 = {𝑥 ∈ ℝ⁺ ∣ (𝑥↑2) ≤ 𝐴}
sqrlem1.2	⊢ 𝐵 = sup(𝑆, ℝ, < )

**Assertion**
Ref	Expression
sqrlem4	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐵 ∈ ℝ⁺ ∧ 𝐵 ≤ 1))

Distinct variable group: 𝑥,𝐴 Allowed substitution hints: 𝐵(𝑥) 𝑆(𝑥)

Proof of Theorem sqrlem4 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sqrlem1.2	. . . 4 ⊢ 𝐵 = sup(𝑆, ℝ, < )
2		sqrlem1.1	. . . . . 6 ⊢ 𝑆 = {𝑥 ∈ ℝ⁺ ∣ (𝑥↑2) ≤ 𝐴}
3	2, 1	sqrlem3 14604	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑦))
4		suprcl 11601	. . . . 5 ⊢ ((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑦) → sup(𝑆, ℝ, < ) ∈ ℝ)
5	3, 4	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → sup(𝑆, ℝ, < ) ∈ ℝ)
6	1, 5	eqeltrid 2917	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐵 ∈ ℝ)
7		rpgt0 12402	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
8	7	adantr 483	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 0 < 𝐴)
9	2, 1	sqrlem2 14603	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆)
10		suprub 11602	. . . . . 6 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑦) ∧ 𝐴 ∈ 𝑆) → 𝐴 ≤ sup(𝑆, ℝ, < ))
11	3, 9, 10	syl2anc 586	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ≤ sup(𝑆, ℝ, < ))
12	11, 1	breqtrrdi 5108	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ≤ 𝐵)
13		0re 10643	. . . . 5 ⊢ 0 ∈ ℝ
14		rpre 12398	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
15		ltletr 10732	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 𝐴 ≤ 𝐵) → 0 < 𝐵))
16	13, 14, 6, 15	mp3an2ani 1464	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → ((0 < 𝐴 ∧ 𝐴 ≤ 𝐵) → 0 < 𝐵))
17	8, 12, 16	mp2and 697	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 0 < 𝐵)
18	6, 17	elrpd 12429	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐵 ∈ ℝ⁺)
19	2, 1	sqrlem1 14602	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → ∀𝑧 ∈ 𝑆 𝑧 ≤ 1)
20		1re 10641	. . . . 5 ⊢ 1 ∈ ℝ
21		suprleub 11607	. . . . 5 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑦) ∧ 1 ∈ ℝ) → (sup(𝑆, ℝ, < ) ≤ 1 ↔ ∀𝑧 ∈ 𝑆 𝑧 ≤ 1))
22	3, 20, 21	sylancl 588	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (sup(𝑆, ℝ, < ) ≤ 1 ↔ ∀𝑧 ∈ 𝑆 𝑧 ≤ 1))
23	19, 22	mpbird 259	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → sup(𝑆, ℝ, < ) ≤ 1)
24	1, 23	eqbrtrid 5101	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐵 ≤ 1)
25	18, 24	jca 514	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐵 ∈ ℝ⁺ ∧ 𝐵 ≤ 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 (class class class)co 7156 supcsup 8904 ℝcr 10536 0cc0 10537 1c1 10538 < clt 10675 ≤ cle 10676 2c2 11693 ℝ⁺crp 12390 ↑cexp 13430

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-cnex 10593 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 ax-pre-sup 10615

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-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 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-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 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-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 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-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431

This theorem is referenced by: sqrlem5 14606 sqrlem7 14608 01sqrex 14609

W3C validator