xrmaxifle - Intuitionistic Logic Explorer

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Theorem xrmaxifle 11723

Description: An upper bound for {𝐴, 𝐵} in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)

**Assertion**
Ref	Expression
xrmaxifle	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))

**Proof of Theorem xrmaxifle**
Step	Hyp	Ref	Expression
1		pnfge 9953	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
2	1	ad2antrr 488	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → 𝐴 ≤ +∞)
3		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → 𝐵 = +∞)
4	3	iftrued 3589	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = +∞)
5	2, 4	breqtrrd 4090	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
6		xrleid 9964	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
7	6	ad3antrrr 492	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 ≤ 𝐴)
8		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐵 = -∞)
9	8	iftrued 3589	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐴)
10	7, 9	breqtrrd 4090	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 ≤ if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
11	1	ad4antr 494	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 ≤ +∞)
12		simpr 110	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 = +∞)
13	12	iftrued 3589	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = +∞)
14	11, 13	breqtrrd 4090	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 ≤ if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
15		mnfle 9956	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → -∞ ≤ 𝐵)
16	15	ad5antlr 497	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → -∞ ≤ 𝐵)
17		simpr 110	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 = -∞)
18	17	iftrued 3589	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = 𝐵)
19	16, 17, 18	3brtr4d 4094	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 ≤ if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
20		simplr 528	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = +∞)
21		simpr 110	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = -∞)
22		elxr 9940	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
23	22	biimpi 120	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
24	23	ad5antr 496	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
25	20, 21, 24	ecase23d 1365	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ∈ ℝ)
26		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → ¬ 𝐵 = +∞)
27	26	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = +∞)
28		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
29	28	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = -∞)
30		elxr 9940	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
31	30	biimpi 120	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
32	31	ad5antlr 497	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
33	27, 29, 32	ecase23d 1365	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐵 ∈ ℝ)
34		maxle1 11688	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ, < ))
35	25, 33, 34	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ, < ))
36	21	iffalsed 3592	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = sup({𝐴, 𝐵}, ℝ, < ))
37	35, 36	breqtrrd 4090	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ≤ if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
38		xrmnfdc 10007	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)
39		exmiddc 840	. . . . . . . . . 10 ⊢ (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
40	38, 39	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
41	40	ad4antr 494	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
42	19, 37, 41	mpjaodan 802	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ≤ if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
43		simpr 110	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → ¬ 𝐴 = +∞)
44	43	iffalsed 3592	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
45	42, 44	breqtrrd 4090	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ≤ if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
46		xrpnfdc 10006	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = +∞)
47		exmiddc 840	. . . . . . . 8 ⊢ (DECID 𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
48	46, 47	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
49	48	ad3antrrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
50	14, 45, 49	mpjaodan 802	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ≤ if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
51	28	iffalsed 3592	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 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, < ))))
52	50, 51	breqtrrd 4090	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ≤ if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
53		xrmnfdc 10007	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = -∞)
54		exmiddc 840	. . . . . 6 ⊢ (DECID 𝐵 = -∞ → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
55	53, 54	syl 14	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
56	55	ad2antlr 489	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
57	10, 52, 56	mpjaodan 802	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → 𝐴 ≤ if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
58	26	iffalsed 3592	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 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, < )))))
59	57, 58	breqtrrd 4090	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
60		xrpnfdc 10006	. . . 4 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = +∞)
61		exmiddc 840	. . . 4 ⊢ (DECID 𝐵 = +∞ → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
62	60, 61	syl 14	. . 3 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
63	62	adantl 277	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
64	5, 59, 63	mpjaodan 802	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 712 DECID wdc 838 ∨ w3o 982 = wceq 1375 ∈ wcel 2180 ifcif 3582 {cpr 3647 class class class wbr 4062 supcsup 7117 ℝcr 7966 +∞cpnf 8146 -∞cmnf 8147 ℝ^*cxr 8148 < clt 8149 ≤ cle 8150

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-frec 6507 df-sup 7119 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-rp 9818 df-seqfrec 10637 df-exp 10728 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476

This theorem is referenced by: xrmaxiflemval 11727 xrmax1sup 11730

W3C validator