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Theorem xrmaxiflemab 11221
Description: Lemma for xrmaxif 11225. A variation of xrmaxleim 11218- 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 3539 . . 3 ((𝜑𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = +∞)
32, 1eqtr4d 2211 . 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 3542 . . 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 4024 . . . . 5 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < -∞)
10 xrmaxiflemab.a . . . . . . 7 (𝜑𝐴 ∈ ℝ*)
11 nltmnf 9757 . . . . . . 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 3542 . . . . 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 4022 . . . . . . 7 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → +∞ < 𝐵)
20 xrmaxiflemab.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ*)
21 pnfnlt 9756 . . . . . . . . 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 3542 . . . . . . 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 3539 . . . . . . . 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 3542 . . . . . . . . 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 9745 . . . . . . . . . . . . . 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 9745 . . . . . . . . . . . . . 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 8050 . . . . . . . . . 10 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴𝐵)
45 maxleim 11180 . . . . . . . . . 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 2208 . . . . . . . 8 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = 𝐵)
48 xrmnfdc 9812 . . . . . . . . . 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 2208 . . . . . 6 ((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = 𝐵)
54 xrpnfdc 9811 . . . . . . . 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 2208 . . . 4 (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐵)
60 xrmnfdc 9812 . . . . . 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 2208 . 2 ((𝜑 ∧ ¬ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
66 xrpnfdc 9811 . . 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 2146  ifcif 3532  {cpr 3590   class class class wbr 3998  supcsup 6971  cr 7785  +∞cpnf 7963  -∞cmnf 7964  *cxr 7965   < clt 7966  cle 7967
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-pre-ltirr 7898  ax-pre-lttrn 7900  ax-pre-apti 7901
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-cnv 4628  df-iota 5170  df-riota 5821  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972
This theorem is referenced by:  xrmaxiflemlub  11222
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