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Theorem xrmaxiflemlub 11240
Description: Lemma for xrmaxif 11243. A least upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 28-Apr-2023.)
Hypotheses
Ref Expression
xrmaxiflemlub.a (𝜑𝐴 ∈ ℝ*)
xrmaxiflemlub.b (𝜑𝐵 ∈ ℝ*)
xrmaxiflemlub.c (𝜑𝐶 ∈ ℝ*)
xrmaxiflemlub.clt (𝜑𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
Assertion
Ref Expression
xrmaxiflemlub (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))

Proof of Theorem xrmaxiflemlub
StepHypRef Expression
1 xrmaxiflemlub.clt . . 3 (𝜑𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
2 xrmaxiflemlub.c . . . 4 (𝜑𝐶 ∈ ℝ*)
3 xrmaxiflemlub.a . . . . 5 (𝜑𝐴 ∈ ℝ*)
4 xrmaxiflemlub.b . . . . 5 (𝜑𝐵 ∈ ℝ*)
5 xrmaxiflemcl 11237 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ*)
63, 4, 5syl2anc 411 . . . 4 (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ*)
7 xrltso 9783 . . . . 5 < Or ℝ*
8 sowlin 4317 . . . . 5 (( < Or ℝ* ∧ (𝐶 ∈ ℝ* ∧ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ*𝐴 ∈ ℝ*)) → (𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → (𝐶 < 𝐴𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))))
97, 8mpan 424 . . . 4 ((𝐶 ∈ ℝ* ∧ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ*𝐴 ∈ ℝ*) → (𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → (𝐶 < 𝐴𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))))
102, 6, 3, 9syl3anc 1238 . . 3 (𝜑 → (𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → (𝐶 < 𝐴𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))))
111, 10mpd 13 . 2 (𝜑 → (𝐶 < 𝐴𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))))
121adantr 276 . . . . 5 ((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
133adantr 276 . . . . . 6 ((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐴 ∈ ℝ*)
144adantr 276 . . . . . 6 ((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐵 ∈ ℝ*)
15 simplr 528 . . . . . . . . 9 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
16 simpr 110 . . . . . . . . . 10 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐵 = +∞)
1716iftrued 3541 . . . . . . . . 9 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = +∞)
1815, 17breqtrd 4026 . . . . . . . 8 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐴 < +∞)
1918, 16breqtrrd 4028 . . . . . . 7 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
20 simplr 528 . . . . . . . . . . . 12 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
21 simpr 110 . . . . . . . . . . . . 13 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → ¬ 𝐵 = +∞)
2221iffalsed 3544 . . . . . . . . . . . 12 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 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, < )))))
2320, 22breqtrd 4026 . . . . . . . . . . 11 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → 𝐴 < if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
2423adantr 276 . . . . . . . . . 10 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
25 simpr 110 . . . . . . . . . . 11 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐵 = -∞)
2625iftrued 3541 . . . . . . . . . 10 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐴)
2724, 26breqtrd 4026 . . . . . . . . 9 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < 𝐴)
28 xrltnr 9766 . . . . . . . . . . 11 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
293, 28syl 14 . . . . . . . . . 10 (𝜑 → ¬ 𝐴 < 𝐴)
3029ad3antrrr 492 . . . . . . . . 9 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐴)
3127, 30pm2.21dd 620 . . . . . . . 8 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < 𝐵)
3223adantr 276 . . . . . . . . . . . . 13 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 < if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
33 simpr 110 . . . . . . . . . . . . . 14 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
3433iffalsed 3544 . . . . . . . . . . . . 13 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
3532, 34breqtrd 4026 . . . . . . . . . . . 12 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 < if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
3635adantr 276 . . . . . . . . . . 11 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 < if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
37 simpr 110 . . . . . . . . . . . 12 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 = +∞)
3837iftrued 3541 . . . . . . . . . . 11 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = +∞)
3936, 38breqtrd 4026 . . . . . . . . . 10 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 < +∞)
40 nltpnft 9801 . . . . . . . . . . . . 13 (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
413, 40syl 14 . . . . . . . . . . . 12 (𝜑 → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
4241ad4antr 494 . . . . . . . . . . 11 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
4337, 42mpbid 147 . . . . . . . . . 10 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → ¬ 𝐴 < +∞)
4439, 43pm2.21dd 620 . . . . . . . . 9 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 < 𝐵)
4535adantr 276 . . . . . . . . . . . . 13 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 < if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
46 simpr 110 . . . . . . . . . . . . . 14 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → ¬ 𝐴 = +∞)
4746iffalsed 3544 . . . . . . . . . . . . 13 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
4845, 47breqtrd 4026 . . . . . . . . . . . 12 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 < if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
4948adantr 276 . . . . . . . . . . 11 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 < if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
50 simpr 110 . . . . . . . . . . . 12 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 = -∞)
5150iftrued 3541 . . . . . . . . . . 11 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = 𝐵)
5249, 51breqtrd 4026 . . . . . . . . . 10 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
5329ad5antr 496 . . . . . . . . . . 11 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 < 𝐴)
54 simp-5l 543 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝜑)
55 simp-4r 542 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = +∞)
5654, 55jca 306 . . . . . . . . . . . . . . . 16 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝜑 ∧ ¬ 𝐵 = +∞))
57 simpllr 534 . . . . . . . . . . . . . . . 16 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = -∞)
5856, 57jca 306 . . . . . . . . . . . . . . 15 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞))
59 simplr 528 . . . . . . . . . . . . . . 15 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = +∞)
6058, 59jca 306 . . . . . . . . . . . . . 14 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞))
61 simpr 110 . . . . . . . . . . . . . 14 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = -∞)
62 simplr 528 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = +∞)
63 simpr 110 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = -∞)
64 elxr 9763 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
653, 64sylib 122 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
6665ad4antr 494 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
6762, 63, 66ecase23d 1350 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ∈ ℝ)
6860, 61, 67syl2anc 411 . . . . . . . . . . . . 13 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ∈ ℝ)
69 simp-4r 542 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = +∞)
70 simpllr 534 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = -∞)
71 elxr 9763 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
724, 71sylib 122 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
7372ad4antr 494 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
7469, 70, 73ecase23d 1350 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐵 ∈ ℝ)
7560, 61, 74syl2anc 411 . . . . . . . . . . . . 13 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐵 ∈ ℝ)
7648adantr 276 . . . . . . . . . . . . . 14 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 < if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
7761iffalsed 3544 . . . . . . . . . . . . . 14 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = sup({𝐴, 𝐵}, ℝ, < ))
7876, 77breqtrd 4026 . . . . . . . . . . . . 13 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 < sup({𝐴, 𝐵}, ℝ, < ))
79 maxleastlt 11208 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ∈ ℝ ∧ 𝐴 < sup({𝐴, 𝐵}, ℝ, < ))) → (𝐴 < 𝐴𝐴 < 𝐵))
8068, 75, 68, 78, 79syl22anc 1239 . . . . . . . . . . . 12 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 < 𝐴𝐴 < 𝐵))
8180orcomd 729 . . . . . . . . . . 11 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 < 𝐵𝐴 < 𝐴))
8253, 81ecased 1349 . . . . . . . . . 10 ((((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 < 𝐵)
83 xrmnfdc 9830 . . . . . . . . . . . 12 (𝐴 ∈ ℝ*DECID 𝐴 = -∞)
84 exmiddc 836 . . . . . . . . . . . 12 (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
853, 83, 843syl 17 . . . . . . . . . . 11 (𝜑 → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
8685ad4antr 494 . . . . . . . . . 10 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
8752, 82, 86mpjaodan 798 . . . . . . . . 9 (((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 < 𝐵)
88 xrpnfdc 9829 . . . . . . . . . . 11 (𝐴 ∈ ℝ*DECID 𝐴 = +∞)
89 exmiddc 836 . . . . . . . . . . 11 (DECID 𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
903, 88, 893syl 17 . . . . . . . . . 10 (𝜑 → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
9190ad3antrrr 492 . . . . . . . . 9 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
9244, 87, 91mpjaodan 798 . . . . . . . 8 ((((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 < 𝐵)
93 xrmnfdc 9830 . . . . . . . . . 10 (𝐵 ∈ ℝ*DECID 𝐵 = -∞)
94 exmiddc 836 . . . . . . . . . 10 (DECID 𝐵 = -∞ → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
954, 93, 943syl 17 . . . . . . . . 9 (𝜑 → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
9695ad2antrr 488 . . . . . . . 8 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
9731, 92, 96mpjaodan 798 . . . . . . 7 (((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → 𝐴 < 𝐵)
98 xrpnfdc 9829 . . . . . . . 8 (𝐵 ∈ ℝ*DECID 𝐵 = +∞)
99 exmiddc 836 . . . . . . . 8 (DECID 𝐵 = +∞ → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
10014, 98, 993syl 17 . . . . . . 7 ((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
10119, 97, 100mpjaodan 798 . . . . . 6 ((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐴 < 𝐵)
10213, 14, 101xrmaxiflemab 11239 . . . . 5 ((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
10312, 102breqtrd 4026 . . . 4 ((𝜑𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐶 < 𝐵)
104103ex 115 . . 3 (𝜑 → (𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → 𝐶 < 𝐵))
105104orim2d 788 . 2 (𝜑 → ((𝐶 < 𝐴𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → (𝐶 < 𝐴𝐶 < 𝐵)))
10611, 105mpd 13 1 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3o 977  w3a 978   = wceq 1353  wcel 2148  ifcif 3534  {cpr 3592   class class class wbr 4000   Or wor 4292  supcsup 6975  cr 7801  +∞cpnf 7979  -∞cmnf 7980  *cxr 7981   < clt 7982
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-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992
This theorem is referenced by:  xrmaxiflemval  11242  xrmaxleastlt  11248
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