Theorem infglb 9458

Description: An infimum is the greatest lower bound. See also infcl 9456 and inflb 9457. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infcl.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

Assertion

Ref	Expression
infglb	⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑧,𝐶   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐶(𝑥, 𝑦)

Proof of Theorem infglb

Step	Hyp	Ref	Expression
1		df-inf 9410	. . . . 5 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2	1	breq1i 5118	. . . 4 ⊢ (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶)
3		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
4		infcl.1	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Or 𝐴)
5		cnvso 6293	. . . . . . . 8 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
6	4, 5	sylib 221	. . . . . . 7 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
7		infcl.2	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
8	4, 7	infcllem 9455	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
9	6, 8	supcl 9425	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
11		brcnvg 5867	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶))
12	11	bicomd 226	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
13	3, 10, 12	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
14	2, 13	bitrid 286	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
15	6, 8	suplub 9427	. . . . 5 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)) → ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧))
16	15	expdimp 458	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) → ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧))
17		vex 3461	. . . . . 6 ⊢ 𝑧 ∈ V
18		brcnvg 5867	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
19	3, 17, 18	sylancl 598	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
20	19	rexbidv 3191	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
21	16, 20	sylibd 242	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
22	14, 21	sylbid 243	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
23	22	expimpd 459	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   class class class wbr 5111   Or wor 5570  ^◡ccnv 5662  supcsup 9407  infcinf 9408

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-po 5571  df-so 5572  df-cnv 5671  df-iota 6496  df-riota 7376  df-sup 9409  df-inf 9410

This theorem is used by:  infnlb  9460  omssubaddlem  34754  omssubadd  34755  gtinf  36887  infxrunb2  46141