Theorem infglb 9491

Description: An infimum is the greatest lower bound. See also infcl 9489 and inflb 9490. (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 9444	. . . . 5 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2	1	breq1i 5155	. . . 4 ⊢ (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶)
3		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
4		infcl.1	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Or 𝐴)
5		cnvso 6287	. . . . . . . 8 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
6	4, 5	sylib 217	. . . . . . 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 9488	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
9	6, 8	supcl 9459	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
11		brcnvg 5879	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶))
12	11	bicomd 222	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
13	3, 10, 12	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
14	2, 13	bitrid 283	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
15	6, 8	suplub 9461	. . . . 5 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)) → ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧))
16	15	expdimp 452	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) → ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧))
17		vex 3477	. . . . . 6 ⊢ 𝑧 ∈ V
18		brcnvg 5879	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
19	3, 17, 18	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
20	19	rexbidv 3177	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
21	16, 20	sylibd 238	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
22	14, 21	sylbid 239	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
23	22	expimpd 453	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  Vcvv 3473   class class class wbr 5148   Or wor 5587  ^◡ccnv 5675  supcsup 9441  infcinf 9442

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-po 5588  df-so 5589  df-cnv 5684  df-iota 6495  df-riota 7368  df-sup 9443  df-inf 9444

This theorem is referenced by:  infnlb  9493  omssubaddlem  33762  omssubadd  33763  gtinf  35668  infxrunb2  44537