Theorem infglb 9398

Description: An infimum is the greatest lower bound. See also infcl 9396 and inflb 9397. (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 9350	. . . . 5 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2	1	breq1i 5093	. . . 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 6247	. . . . . . . 8 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
6	4, 5	sylib 218	. . . . . . 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 9395	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
9	6, 8	supcl 9365	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
11		brcnvg 5829	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶))
12	11	bicomd 223	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
13	3, 10, 12	syl2anc 585	. . . 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 9367	. . . . 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 3434	. . . . . 6 ⊢ 𝑧 ∈ V
18		brcnvg 5829	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
19	3, 17, 18	sylancl 587	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
20	19	rexbidv 3162	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
21	16, 20	sylibd 239	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
22	14, 21	sylbid 240	. 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 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   class class class wbr 5086   Or wor 5532  ^◡ccnv 5624  supcsup 9347  infcinf 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-po 5533  df-so 5534  df-cnv 5633  df-iota 6449  df-riota 7318  df-sup 9349  df-inf 9350

This theorem is referenced by:  infnlb  9400  omssubaddlem  34462  omssubadd  34463  gtinf  36520  infxrunb2  45818