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Theorem rexanre 14036
Description: Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
Assertion
Ref Expression
rexanre (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓))))
Distinct variable groups:   𝑗,𝑘,𝐴   𝜑,𝑗   𝜓,𝑗
Allowed substitution hints:   𝜑(𝑘)   𝜓(𝑘)

Proof of Theorem rexanre
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 473 . . . . . 6 ((𝜑𝜓) → 𝜑)
21imim2i 16 . . . . 5 ((𝑗𝑘 → (𝜑𝜓)) → (𝑗𝑘𝜑))
32ralimi 2948 . . . 4 (∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) → ∀𝑘𝐴 (𝑗𝑘𝜑))
43reximi 3007 . . 3 (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑))
5 simpr 477 . . . . . 6 ((𝜑𝜓) → 𝜓)
65imim2i 16 . . . . 5 ((𝑗𝑘 → (𝜑𝜓)) → (𝑗𝑘𝜓))
76ralimi 2948 . . . 4 (∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) → ∀𝑘𝐴 (𝑗𝑘𝜓))
87reximi 3007 . . 3 (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓))
94, 8jca 554 . 2 (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) → (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓)))
10 breq1 4626 . . . . . . . 8 (𝑗 = 𝑥 → (𝑗𝑘𝑥𝑘))
1110imbi1d 331 . . . . . . 7 (𝑗 = 𝑥 → ((𝑗𝑘𝜑) ↔ (𝑥𝑘𝜑)))
1211ralbidv 2982 . . . . . 6 (𝑗 = 𝑥 → (∀𝑘𝐴 (𝑗𝑘𝜑) ↔ ∀𝑘𝐴 (𝑥𝑘𝜑)))
1312cbvrexv 3164 . . . . 5 (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝐴 (𝑥𝑘𝜑))
14 breq1 4626 . . . . . . . 8 (𝑗 = 𝑦 → (𝑗𝑘𝑦𝑘))
1514imbi1d 331 . . . . . . 7 (𝑗 = 𝑦 → ((𝑗𝑘𝜓) ↔ (𝑦𝑘𝜓)))
1615ralbidv 2982 . . . . . 6 (𝑗 = 𝑦 → (∀𝑘𝐴 (𝑗𝑘𝜓) ↔ ∀𝑘𝐴 (𝑦𝑘𝜓)))
1716cbvrexv 3164 . . . . 5 (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓) ↔ ∃𝑦 ∈ ℝ ∀𝑘𝐴 (𝑦𝑘𝜓))
1813, 17anbi12i 732 . . . 4 ((∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓)) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∃𝑦 ∈ ℝ ∀𝑘𝐴 (𝑦𝑘𝜓)))
19 reeanv 3101 . . . 4 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∀𝑘𝐴 (𝑦𝑘𝜓)) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∃𝑦 ∈ ℝ ∀𝑘𝐴 (𝑦𝑘𝜓)))
2018, 19bitr4i 267 . . 3 ((∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∀𝑘𝐴 (𝑦𝑘𝜓)))
21 ifcl 4108 . . . . . . 7 ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → if(𝑥𝑦, 𝑦, 𝑥) ∈ ℝ)
2221ancoms 469 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → if(𝑥𝑦, 𝑦, 𝑥) ∈ ℝ)
2322adantl 482 . . . . 5 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → if(𝑥𝑦, 𝑦, 𝑥) ∈ ℝ)
24 r19.26 3059 . . . . . 6 (∀𝑘𝐴 ((𝑥𝑘𝜑) ∧ (𝑦𝑘𝜓)) ↔ (∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∀𝑘𝐴 (𝑦𝑘𝜓)))
25 prth 594 . . . . . . . 8 (((𝑥𝑘𝜑) ∧ (𝑦𝑘𝜓)) → ((𝑥𝑘𝑦𝑘) → (𝜑𝜓)))
26 simplrl 799 . . . . . . . . . 10 (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘𝐴) → 𝑥 ∈ ℝ)
27 simplrr 800 . . . . . . . . . 10 (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘𝐴) → 𝑦 ∈ ℝ)
28 simpl 473 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝐴 ⊆ ℝ)
2928sselda 3588 . . . . . . . . . 10 (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘𝐴) → 𝑘 ∈ ℝ)
30 maxle 11981 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 ↔ (𝑥𝑘𝑦𝑘)))
3126, 27, 29, 30syl3anc 1323 . . . . . . . . 9 (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘𝐴) → (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 ↔ (𝑥𝑘𝑦𝑘)))
3231imbi1d 331 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘𝐴) → ((if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑𝜓)) ↔ ((𝑥𝑘𝑦𝑘) → (𝜑𝜓))))
3325, 32syl5ibr 236 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘𝐴) → (((𝑥𝑘𝜑) ∧ (𝑦𝑘𝜓)) → (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑𝜓))))
3433ralimdva 2958 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (∀𝑘𝐴 ((𝑥𝑘𝜑) ∧ (𝑦𝑘𝜓)) → ∀𝑘𝐴 (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑𝜓))))
3524, 34syl5bir 233 . . . . 5 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∀𝑘𝐴 (𝑦𝑘𝜓)) → ∀𝑘𝐴 (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑𝜓))))
36 breq1 4626 . . . . . . . 8 (𝑗 = if(𝑥𝑦, 𝑦, 𝑥) → (𝑗𝑘 ↔ if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘))
3736imbi1d 331 . . . . . . 7 (𝑗 = if(𝑥𝑦, 𝑦, 𝑥) → ((𝑗𝑘 → (𝜑𝜓)) ↔ (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑𝜓))))
3837ralbidv 2982 . . . . . 6 (𝑗 = if(𝑥𝑦, 𝑦, 𝑥) → (∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) ↔ ∀𝑘𝐴 (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑𝜓))))
3938rspcev 3299 . . . . 5 ((if(𝑥𝑦, 𝑦, 𝑥) ∈ ℝ ∧ ∀𝑘𝐴 (if(𝑥𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑𝜓))) → ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)))
4023, 35, 39syl6an 567 . . . 4 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∀𝑘𝐴 (𝑦𝑘𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓))))
4140rexlimdvva 3033 . . 3 (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (∀𝑘𝐴 (𝑥𝑘𝜑) ∧ ∀𝑘𝐴 (𝑦𝑘𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓))))
4220, 41syl5bi 232 . 2 (𝐴 ⊆ ℝ → ((∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓))))
439, 42impbid2 216 1 (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  wss 3560  ifcif 4064   class class class wbr 4623  cr 9895  cle 10035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-pre-lttri 9970  ax-pre-lttrn 9971
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040
This theorem is referenced by:  o1lo1  14218  rlimuni  14231  lo1add  14307  lo1mul  14308  rlimno1  14334
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