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Theorem xrmaxiflemab 11239
Description: Lemma for xrmaxif 11243. A variation of xrmaxleim 11236- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)
Hypotheses
Ref Expression
xrmaxiflemab.a (𝜑𝐴 ∈ ℝ*)
xrmaxiflemab.b (𝜑𝐵 ∈ ℝ*)
xrmaxiflemab.ab (𝜑𝐴 < 𝐵)
Assertion
Ref Expression
xrmaxiflemab (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)

Proof of Theorem xrmaxiflemab
StepHypRef Expression
1 simpr 110 . . . 4 ((𝜑𝐵 = +∞) → 𝐵 = +∞)
21iftrued 3541 . . 3 ((𝜑𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = +∞)
32, 1eqtr4d 2213 . 2 ((𝜑𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
4 simpr 110 . . . 4 ((𝜑 ∧ ¬ 𝐵 = +∞) → ¬ 𝐵 = +∞)
54iffalsed 3544 . . 3 ((𝜑 ∧ ¬ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
6 xrmaxiflemab.ab . . . . . . 7 (𝜑𝐴 < 𝐵)
76ad2antrr 488 . . . . . 6 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < 𝐵)
8 simpr 110 . . . . . 6 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐵 = -∞)
97, 8breqtrd 4026 . . . . 5 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < -∞)
10 xrmaxiflemab.a . . . . . . 7 (𝜑𝐴 ∈ ℝ*)
11 nltmnf 9775 . . . . . . 7 (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
1210, 11syl 14 . . . . . 6 (𝜑 → ¬ 𝐴 < -∞)
1312ad2antrr 488 . . . . 5 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
149, 13pm2.21dd 620 . . . 4 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐵)
15 simpr 110 . . . . . 6 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
1615iffalsed 3544 . . . . 5 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
17 simpr 110 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 = +∞)
186ad3antrrr 492 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 < 𝐵)
1917, 18eqbrtrrd 4024 . . . . . . 7 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → +∞ < 𝐵)
20 xrmaxiflemab.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ*)
21 pnfnlt 9774 . . . . . . . . 9 (𝐵 ∈ ℝ* → ¬ +∞ < 𝐵)
2220, 21syl 14 . . . . . . . 8 (𝜑 → ¬ +∞ < 𝐵)
2322ad3antrrr 492 . . . . . . 7 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → ¬ +∞ < 𝐵)
2419, 23pm2.21dd 620 . . . . . 6 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = 𝐵)
25 simpr 110 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → ¬ 𝐴 = +∞)
2625iffalsed 3544 . . . . . . 7 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
27 simpr 110 . . . . . . . . 9 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 = -∞)
2827iftrued 3541 . . . . . . . 8 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = 𝐵)
29 simpr 110 . . . . . . . . . 10 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = -∞)
3029iffalsed 3544 . . . . . . . . 9 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = sup({𝐴, 𝐵}, ℝ, < ))
3125adantr 276 . . . . . . . . . . . 12 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = +∞)
32 elxr 9763 . . . . . . . . . . . . . 14 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3310, 32sylib 122 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3433ad4antr 494 . . . . . . . . . . . 12 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3531, 29, 34ecase23d 1350 . . . . . . . . . . 11 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ∈ ℝ)
364ad3antrrr 492 . . . . . . . . . . . 12 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = +∞)
3715ad2antrr 488 . . . . . . . . . . . 12 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = -∞)
38 elxr 9763 . . . . . . . . . . . . . 14 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3920, 38sylib 122 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
4039ad4antr 494 . . . . . . . . . . . 12 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
4136, 37, 40ecase23d 1350 . . . . . . . . . . 11 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐵 ∈ ℝ)
4235, 41jca 306 . . . . . . . . . 10 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
436ad4antr 494 . . . . . . . . . . 11 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 < 𝐵)
4435, 41, 43ltled 8066 . . . . . . . . . 10 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴𝐵)
45 maxleim 11198 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 → sup({𝐴, 𝐵}, ℝ, < ) = 𝐵))
4642, 44, 45sylc 62 . . . . . . . . 9 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → sup({𝐴, 𝐵}, ℝ, < ) = 𝐵)
4730, 46eqtrd 2210 . . . . . . . 8 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = 𝐵)
48 xrmnfdc 9830 . . . . . . . . . 10 (𝐴 ∈ ℝ*DECID 𝐴 = -∞)
49 exmiddc 836 . . . . . . . . . 10 (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
5010, 48, 493syl 17 . . . . . . . . 9 (𝜑 → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
5150ad3antrrr 492 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
5228, 47, 51mpjaodan 798 . . . . . . 7 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = 𝐵)
5326, 52eqtrd 2210 . . . . . 6 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = 𝐵)
54 xrpnfdc 9829 . . . . . . . 8 (𝐴 ∈ ℝ*DECID 𝐴 = +∞)
55 exmiddc 836 . . . . . . . 8 (DECID 𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5610, 54, 553syl 17 . . . . . . 7 (𝜑 → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5756ad2antrr 488 . . . . . 6 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5824, 53, 57mpjaodan 798 . . . . 5 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = 𝐵)
5916, 58eqtrd 2210 . . . 4 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐵)
60 xrmnfdc 9830 . . . . . 6 (𝐵 ∈ ℝ*DECID 𝐵 = -∞)
61 exmiddc 836 . . . . . 6 (DECID 𝐵 = -∞ → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
6220, 60, 613syl 17 . . . . 5 (𝜑 → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
6362adantr 276 . . . 4 ((𝜑 ∧ ¬ 𝐵 = +∞) → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
6414, 59, 63mpjaodan 798 . . 3 ((𝜑 ∧ ¬ 𝐵 = +∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐵)
655, 64eqtrd 2210 . 2 ((𝜑 ∧ ¬ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
66 xrpnfdc 9829 . . 3 (𝐵 ∈ ℝ*DECID 𝐵 = +∞)
67 exmiddc 836 . . 3 (DECID 𝐵 = +∞ → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
6820, 66, 673syl 17 . 2 (𝜑 → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
693, 65, 68mpjaodan 798 1 (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834  w3o 977   = wceq 1353  wcel 2148  ifcif 3534  {cpr 3592   class class class wbr 4000  supcsup 6975  cr 7801  +∞cpnf 7979  -∞cmnf 7980  *cxr 7981   < clt 7982  cle 7983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-pre-ltirr 7914  ax-pre-lttrn 7916  ax-pre-apti 7917
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4629  df-cnv 4631  df-iota 5174  df-riota 5825  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988
This theorem is referenced by:  xrmaxiflemlub  11240
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