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Theorem xrmaxiflemcom 11998

Description: Lemma for xrmaxif 12000. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)

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

**Proof of Theorem xrmaxiflemcom**
Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → 𝐵 = +∞)
2	1	iftrued 3647	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = +∞)
3		xrpnfdc 10227	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = +∞)
4	3	ad2antrr 492	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → DECID 𝐴 = +∞)
5		exmiddc 848	. . . . . 6 ⊢ (DECID 𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
6	4, 5	syl 14	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
7		eqid 2238	. . . . . . . 8 ⊢ +∞ = +∞
8	7	biantru 302	. . . . . . 7 ⊢ (𝐴 = +∞ ↔ (𝐴 = +∞ ∧ +∞ = +∞))
9	8	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (𝐴 = +∞ ↔ (𝐴 = +∞ ∧ +∞ = +∞)))
10		xrmnfdc 10228	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)
11	10	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → DECID 𝐴 = -∞)
12		exmiddc 848	. . . . . . . . . 10 ⊢ (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
13	11, 12	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
14		iba 300	. . . . . . . . . . . 12 ⊢ (+∞ = 𝐵 → (𝐴 = -∞ ↔ (𝐴 = -∞ ∧ +∞ = 𝐵)))
15	14	eqcoms 2241	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (𝐴 = -∞ ↔ (𝐴 = -∞ ∧ +∞ = 𝐵)))
16	15	adantl 277	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (𝐴 = -∞ ↔ (𝐴 = -∞ ∧ +∞ = 𝐵)))
17	1	iftrued 3647	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))) = +∞)
18	17	eqcomd 2244	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → +∞ = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))
19	18	biantrud 304	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (¬ 𝐴 = -∞ ↔ (¬ 𝐴 = -∞ ∧ +∞ = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))
20	16, 19	orbi12d 805	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → ((𝐴 = -∞ ∨ ¬ 𝐴 = -∞) ↔ ((𝐴 = -∞ ∧ +∞ = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ +∞ = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))))
21	13, 20	mpbid 147	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → ((𝐴 = -∞ ∧ +∞ = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ +∞ = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))
22		eqifdc 3677	. . . . . . . . 9 ⊢ (DECID 𝐴 = -∞ → (+∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))) ↔ ((𝐴 = -∞ ∧ +∞ = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ +∞ = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))))
23	11, 22	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (+∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))) ↔ ((𝐴 = -∞ ∧ +∞ = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ +∞ = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))))
24	21, 23	mpbird 167	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → +∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))
25	24	biantrud 304	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (¬ 𝐴 = +∞ ↔ (¬ 𝐴 = +∞ ∧ +∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))))
26	9, 25	orbi12d 805	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → ((𝐴 = +∞ ∨ ¬ 𝐴 = +∞) ↔ ((𝐴 = +∞ ∧ +∞ = +∞) ∨ (¬ 𝐴 = +∞ ∧ +∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))))
27	6, 26	mpbid 147	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → ((𝐴 = +∞ ∧ +∞ = +∞) ∨ (¬ 𝐴 = +∞ ∧ +∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))))
28		eqifdc 3677	. . . . 5 ⊢ (DECID 𝐴 = +∞ → (+∞ = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))) ↔ ((𝐴 = +∞ ∧ +∞ = +∞) ∨ (¬ 𝐴 = +∞ ∧ +∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))))
29	4, 28	syl 14	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → (+∞ = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))) ↔ ((𝐴 = +∞ ∧ +∞ = +∞) ∨ (¬ 𝐴 = +∞ ∧ +∞ = if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))))
30	27, 29	mpbird 167	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → +∞ = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))
31	2, 30	eqtrd 2271	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))
32	3, 5	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
33	32	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
34		pm4.24 399	. . . . . . . . 9 ⊢ (𝐴 = +∞ ↔ (𝐴 = +∞ ∧ 𝐴 = +∞))
35	34	a1i 9	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (𝐴 = +∞ ↔ (𝐴 = +∞ ∧ 𝐴 = +∞)))
36	10	ad3antrrr 496	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → DECID 𝐴 = -∞)
37	36, 12	syl 14	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
38		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐵 = -∞)
39	38	eqeq2d 2250	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ↔ 𝐴 = -∞))
40	39	anbi2d 468	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ((𝐴 = -∞ ∧ 𝐴 = 𝐵) ↔ (𝐴 = -∞ ∧ 𝐴 = -∞)))
41		anidm 400	. . . . . . . . . . . . 13 ⊢ ((𝐴 = -∞ ∧ 𝐴 = -∞) ↔ 𝐴 = -∞)
42	40, 41	bitr2di 197	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (𝐴 = -∞ ↔ (𝐴 = -∞ ∧ 𝐴 = 𝐵)))
43		eqid 2238	. . . . . . . . . . . . . 14 ⊢ 𝐴 = 𝐴
44	43	biantru 302	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 = -∞ ↔ (¬ 𝐴 = -∞ ∧ 𝐴 = 𝐴))
45	44	a1i 9	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (¬ 𝐴 = -∞ ↔ (¬ 𝐴 = -∞ ∧ 𝐴 = 𝐴)))
46	42, 45	orbi12d 805	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ((𝐴 = -∞ ∨ ¬ 𝐴 = -∞) ↔ ((𝐴 = -∞ ∧ 𝐴 = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ 𝐴 = 𝐴))))
47	37, 46	mpbid 147	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ((𝐴 = -∞ ∧ 𝐴 = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ 𝐴 = 𝐴)))
48		eqifdc 3677	. . . . . . . . . . 11 ⊢ (DECID 𝐴 = -∞ → (𝐴 = if(𝐴 = -∞, 𝐵, 𝐴) ↔ ((𝐴 = -∞ ∧ 𝐴 = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ 𝐴 = 𝐴))))
49	36, 48	syl 14	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (𝐴 = if(𝐴 = -∞, 𝐵, 𝐴) ↔ ((𝐴 = -∞ ∧ 𝐴 = 𝐵) ∨ (¬ 𝐴 = -∞ ∧ 𝐴 = 𝐴))))
50	47, 49	mpbird 167	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 = if(𝐴 = -∞, 𝐵, 𝐴))
51	50	biantrud 304	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (¬ 𝐴 = +∞ ↔ (¬ 𝐴 = +∞ ∧ 𝐴 = if(𝐴 = -∞, 𝐵, 𝐴))))
52	35, 51	orbi12d 805	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ((𝐴 = +∞ ∨ ¬ 𝐴 = +∞) ↔ ((𝐴 = +∞ ∧ 𝐴 = +∞) ∨ (¬ 𝐴 = +∞ ∧ 𝐴 = if(𝐴 = -∞, 𝐵, 𝐴)))))
53	33, 52	mpbid 147	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ((𝐴 = +∞ ∧ 𝐴 = +∞) ∨ (¬ 𝐴 = +∞ ∧ 𝐴 = if(𝐴 = -∞, 𝐵, 𝐴))))
54	3	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → DECID 𝐴 = +∞)
55		eqifdc 3677	. . . . . . 7 ⊢ (DECID 𝐴 = +∞ → (𝐴 = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, 𝐴)) ↔ ((𝐴 = +∞ ∧ 𝐴 = +∞) ∨ (¬ 𝐴 = +∞ ∧ 𝐴 = if(𝐴 = -∞, 𝐵, 𝐴)))))
56	54, 55	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → (𝐴 = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, 𝐴)) ↔ ((𝐴 = +∞ ∧ 𝐴 = +∞) ∨ (¬ 𝐴 = +∞ ∧ 𝐴 = if(𝐴 = -∞, 𝐵, 𝐴)))))
57	53, 56	mpbird 167	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, 𝐴)))
58	38	iftrued 3647	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐴)
59	38	iftrued 3647	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )) = 𝐴)
60	59	ifeq2d 3659	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))) = if(𝐴 = -∞, 𝐵, 𝐴))
61	60	ifeq2d 3659	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, 𝐴)))
62	57, 58, 61	3eqtr4d 2281	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))
63		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
64	63	iffalsed 3650	. . . . 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, < ))))
65	63	iffalsed 3650	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )) = sup({𝐵, 𝐴}, ℝ, < ))
66	65	ifeq2d 3659	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))) = if(𝐴 = -∞, 𝐵, sup({𝐵, 𝐴}, ℝ, < )))
67	66	ifeq2d 3659	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐵, 𝐴}, ℝ, < ))))
68		maxcom 11952	. . . . . . 7 ⊢ sup({𝐵, 𝐴}, ℝ, < ) = sup({𝐴, 𝐵}, ℝ, < )
69		ifeq2 3644	. . . . . . 7 ⊢ (sup({𝐵, 𝐴}, ℝ, < ) = sup({𝐴, 𝐵}, ℝ, < ) → if(𝐴 = -∞, 𝐵, sup({𝐵, 𝐴}, ℝ, < )) = if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
70		ifeq2 3644	. . . . . . 7 ⊢ (if(𝐴 = -∞, 𝐵, sup({𝐵, 𝐴}, ℝ, < )) = if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐵, 𝐴}, ℝ, < ))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
71	68, 69, 70	mp2b 8	. . . . . 6 ⊢ if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐵, 𝐴}, ℝ, < ))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
72	67, 71	eqtrdi 2287	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
73	64, 72	eqtr4d 2274	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))
74		xrmnfdc 10228	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = -∞)
75		exmiddc 848	. . . . . 6 ⊢ (DECID 𝐵 = -∞ → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
76	74, 75	syl 14	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
77	76	ad2antlr 493	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
78	62, 73, 77	mpjaodan 810	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))
79		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → ¬ 𝐵 = +∞)
80	79	iffalsed 3650	. . 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, < )))))
81	79	iffalsed 3650	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))) = if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))
82	81	ifeq2d 3659	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))) = if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))
83	82	ifeq2d 3659	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))
84	78, 80, 83	3eqtr4d 2281	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))
85		xrpnfdc 10227	. . . 4 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = +∞)
86		exmiddc 848	. . . 4 ⊢ (DECID 𝐵 = +∞ → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
87	85, 86	syl 14	. . 3 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
88	87	adantl 277	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
89	31, 84, 88	mpjaodan 810	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ifcif 3638 {cpr 3709 supcsup 7316 ℝcr 8172 +∞cpnf 8351 -∞cmnf 8352 ℝ^*cxr 8353 < clt 8354

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-un 4576 ax-setind 4682 ax-cnex 8264 ax-resscn 8265

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-uni 3934 df-sup 7318 df-pnf 8356 df-mnf 8357 df-xr 8358

This theorem is referenced by: xrmaxiflemval 11999

W3C validator