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Theorem supminfrnmpt 43766
Description: The indexed supremum of a bounded-above set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
supminfrnmpt.x 𝑥𝜑
supminfrnmpt.a (𝜑𝐴 ≠ ∅)
supminfrnmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
supminfrnmpt.y (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
Assertion
Ref Expression
supminfrnmpt (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem supminfrnmpt
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supminfrnmpt.x . . . 4 𝑥𝜑
2 eqid 2733 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 43504 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 supminfrnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 6197 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 supminfrnmpt.y . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
81, 7rnmptbdd 43559 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
9 supminf 12865 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ ∧ ran (𝑥𝐴𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
104, 6, 8, 9syl3anc 1372 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
11 eqid 2733 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
12 simpr 486 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → -𝑤 ∈ ran (𝑥𝐴𝐵))
13 renegcl 11469 . . . . . . . . . . . . . 14 (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
142elrnmpt 5912 . . . . . . . . . . . . . 14 (-𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1513, 14syl 17 . . . . . . . . . . . . 13 (𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1615adantr 482 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1712, 16mpbid 231 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
1817adantll 713 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
19 nfv 1918 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
201, 19nfan 1903 . . . . . . . . . . . 12 𝑥(𝜑𝑤 ∈ ℝ)
21 negeq 11398 . . . . . . . . . . . . . . . . . . 19 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
2221eqcomd 2739 . . . . . . . . . . . . . . . . . 18 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
2322adantl 483 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
24 recn 11146 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
2524negnegd 11508 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
2625adantr 482 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
2723, 26eqtr2d 2774 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
2827ex 414 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → (-𝑤 = 𝐵𝑤 = -𝐵))
2928adantl 483 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ℝ) → (-𝑤 = 𝐵𝑤 = -𝐵))
3029adantr 482 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
31 negeq 11398 . . . . . . . . . . . . . . . . 17 (𝑤 = -𝐵 → -𝑤 = --𝐵)
3231adantl 483 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
333recnd 11188 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
3433negnegd 11508 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
3534adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
3632, 35eqtrd 2773 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
3736ex 414 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
3837adantlr 714 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
3930, 38impbid 211 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
4020, 39rexbida 3254 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℝ) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4140adantr 482 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4218, 41mpbid 231 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
43 simplr 768 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ℝ)
4411, 42, 43elrnmptd 5917 . . . . . . . 8 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4544ex 414 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4645ralrimiva 3140 . . . . . 6 (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
47 rabss 4030 . . . . . 6 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4846, 47sylibr 233 . . . . 5 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵))
49 nfcv 2904 . . . . . . . 8 𝑥-𝑤
50 nfmpt1 5214 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
5150nfrn 5908 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
5249, 51nfel 2918 . . . . . . 7 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
53 nfcv 2904 . . . . . . 7 𝑥
5452, 53nfrabw 3439 . . . . . 6 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
5531eleq1d 2819 . . . . . . 7 (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ --𝐵 ∈ ran (𝑥𝐴𝐵)))
563renegcld 11587 . . . . . . 7 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
57 simpr 486 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
582elrnmpt1 5914 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥𝐴𝐵))
5957, 3, 58syl2anc 585 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6034, 59eqeltrd 2834 . . . . . . 7 ((𝜑𝑥𝐴) → --𝐵 ∈ ran (𝑥𝐴𝐵))
6155, 56, 60elrabd 3648 . . . . . 6 ((𝜑𝑥𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
621, 54, 11, 61rnmptssdf 43569 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
6348, 62eqssd 3962 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
6463infeq1d 9418 . . 3 (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6564negeqd 11400 . 2 (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6610, 65eqtrd 2773 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  wne 2940  wral 3061  wrex 3070  {crab 3406  wss 3911  c0 4283   class class class wbr 5106  cmpt 5189  ran crn 5635  supcsup 9381  infcinf 9382  cr 11055   < clt 11194  cle 11195  -cneg 11391
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-inf 9384  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393
This theorem is referenced by: (None)
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