Theorem infxrgelbrnmpt 44151

Description: The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
infxrgelbrnmpt.x	⊢ Ⅎ𝑥𝜑
infxrgelbrnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
infxrgelbrnmpt.c	⊢ (𝜑 → 𝐶 ∈ ℝ*)

Assertion

Ref	Expression
infxrgelbrnmpt	⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem infxrgelbrnmpt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxrgelbrnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2733	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infxrgelbrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 43881	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5		infxrgelbrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
6		infxrgelb 13311	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧))
7	4, 5, 6	syl2anc 585	. 2 ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧))
8		nfmpt1 5256	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	8	nfrn 5950	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
10		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥 𝐶 ≤ 𝑧
11	9, 10	nfralw 3309	. . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧
12	1, 11	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
13		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
14	2	elrnmpt1 5956	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
15	13, 3, 14	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
16	15	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
17		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
18		breq2 5152	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐶 ≤ 𝑧 ↔ 𝐶 ≤ 𝐵))
19	18	rspcva 3611	. . . . . 6 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → 𝐶 ≤ 𝐵)
20	16, 17, 19	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵)
21	20	ex 414	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → (𝑥 ∈ 𝐴 → 𝐶 ≤ 𝐵))
22	12, 21	ralrimi 3255	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵)
23		vex 3479	. . . . . . . . 9 ⊢ 𝑧 ∈ V
24	2	elrnmpt 5954	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
25	23, 24	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
26	25	biimpi 215	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
27	26	adantl 483	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
28		nfra1 3282	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵
29		rspa 3246	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵)
30	18	biimprcd 249	. . . . . . . . . 10 ⊢ (𝐶 ≤ 𝐵 → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
32	31	ex 414	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧)))
33	28, 10, 32	rexlimd 3264	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
34	33	adantr 482	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
35	27, 34	mpd 15	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝐶 ≤ 𝑧)
36	35	ralrimiva 3147	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
37	36	adantl 483	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
38	22, 37	impbida 800	. 2 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧 ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))
39	7, 38	bitrd 279	1 ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  Ⅎwnf 1786   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3948   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677  infcinf 9433  ℝ^*cxr 11244   < clt 11245   ≤ cle 11246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-inf 9435  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444

This theorem is referenced by: (None)