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Theorem xrmaxaddlem 12052

Description: Lemma for xrmaxadd 12053. The case where 𝐴 is real. (Contributed by Jim Kingdon, 11-May-2023.)

**Assertion**
Ref	Expression
xrmaxaddlem	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}, ℝ^, < ) = (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))

Proof of Theorem xrmaxaddlem Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 10217	. . 3 ⊢ ((𝑓 ∈ ℝ^* ∧ 𝑔 ∈ ℝ^*) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2	1	adantl 277	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑓 ∈ ℝ^ ∧ 𝑔 ∈ ℝ^*)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3		rexr 8378	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
4		simp1 1028	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
5		simp2 1029	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
6		simp3 1030	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7		xrmaxcl 12044	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
8	5, 6, 7	syl2anc 415	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
9	4, 8	xaddcld 10304	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
10	3, 9	syl3an1 1311	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
11		elpri 3732	. . . . 5 ⊢ (𝑥 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)} → (𝑥 = (𝐴 +_e 𝐵) ∨ 𝑥 = (𝐴 +_e 𝐶)))
12		simpr 110	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑥 = (𝐴 +_e 𝐵)) → 𝑥 = (𝐴 +_e 𝐵))
13		xrmax1sup 12045	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
14	5, 6, 13	syl2anc 415	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
15		xleadd2a 10294	. . . . . . . . 9 ⊢ (((𝐵 ∈ ℝ^* ∧ sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^ ∧ 𝐴 ∈ ℝ^) ∧ 𝐵 ≤ sup({𝐵, 𝐶}, ℝ^, < )) → (𝐴 +_e 𝐵) ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))
16	5, 8, 4, 14, 15	syl31anc 1281	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_e 𝐵) ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
17	16	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐵)) → (𝐴 +_e 𝐵) ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
18	12, 17	eqbrtrd 4153	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐵)) → 𝑥 ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
19		simpr 110	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑥 = (𝐴 +_e 𝐶)) → 𝑥 = (𝐴 +_e 𝐶))
20		xrmax2sup 12046	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
21	5, 6, 20	syl2anc 415	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
22		xleadd2a 10294	. . . . . . . . 9 ⊢ (((𝐶 ∈ ℝ^* ∧ sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^ ∧ 𝐴 ∈ ℝ^) ∧ 𝐶 ≤ sup({𝐵, 𝐶}, ℝ^, < )) → (𝐴 +_e 𝐶) ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))
23	6, 8, 4, 21, 22	syl31anc 1281	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_e 𝐶) ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
24	23	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐶)) → (𝐴 +_e 𝐶) ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
25	19, 24	eqbrtrd 4153	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐶)) → 𝑥 ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
26	18, 25	jaodan 809	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 = (𝐴 +_e 𝐵) ∨ 𝑥 = (𝐴 +_e 𝐶))) → 𝑥 ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
27	11, 26	sylan2 286	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}) → 𝑥 ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
28	4, 5	xaddcld 10304	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_e 𝐵) ∈ ℝ^)
29	28	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐵)) → (𝐴 +_e 𝐵) ∈ ℝ^)
30	12, 29	eqeltrd 2315	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐵)) → 𝑥 ∈ ℝ^)
31	4, 6	xaddcld 10304	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_e 𝐶) ∈ ℝ^)
32	31	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐶)) → (𝐴 +_e 𝐶) ∈ ℝ^)
33	19, 32	eqeltrd 2315	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_e 𝐶)) → 𝑥 ∈ ℝ^)
34	30, 33	jaodan 809	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 = (𝐴 +_e 𝐵) ∨ 𝑥 = (𝐴 +_e 𝐶))) → 𝑥 ∈ ℝ^)
35	11, 34	sylan2 286	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}) → 𝑥 ∈ ℝ^)
36	9	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}) → (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
37		xrlenlt 8397	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^) → (𝑥 ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ↔ ¬ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) < 𝑥))
38	35, 36, 37	syl2anc 415	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}) → (𝑥 ≤ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ↔ ¬ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )) < 𝑥))
39	27, 38	mpbid 147	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}) → ¬ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) < 𝑥)
40	3, 39	syl3anl1 1326	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}) → ¬ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) < 𝑥)
41	3	3ad2ant1 1049	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
42	41	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → 𝐴 ∈ ℝ^)
43	42	adantr 276	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → 𝐴 ∈ ℝ^)
44		simpl2 1032	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → 𝐵 ∈ ℝ^)
45	44	adantr 276	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → 𝐵 ∈ ℝ^)
46	43, 45	xaddcld 10304	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → (𝐴 +_e 𝐵) ∈ ℝ^)
47		prid1g 3815	. . . . 5 ⊢ ((𝐴 +_e 𝐵) ∈ ℝ^* → (𝐴 +_e 𝐵) ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)})
48	46, 47	syl 14	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → (𝐴 +_e 𝐵) ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)})
49		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → (𝑥 +_e -_e𝐴) < 𝐵)
50		simprl 535	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → 𝑥 ∈ ℝ^)
51	42	xnegcld 10275	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → -_e𝐴 ∈ ℝ^)
52	50, 51	xaddcld 10304	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → (𝑥 +_e -_e𝐴) ∈ ℝ^)
53	52	adantr 276	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → (𝑥 +_e -_e𝐴) ∈ ℝ^)
54		simpl1 1031	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) → 𝐴 ∈ ℝ)
55	54	adantr 276	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → 𝐴 ∈ ℝ)
56		xltadd1 10296	. . . . . . 7 ⊢ (((𝑥 +_e -_e𝐴) ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝑥 +_e -_e𝐴) < 𝐵 ↔ ((𝑥 +_e -_e𝐴) +_e 𝐴) < (𝐵 +_e 𝐴)))
57	53, 45, 55, 56	syl3anc 1278	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → ((𝑥 +_e -_e𝐴) < 𝐵 ↔ ((𝑥 +_e -_e𝐴) +_e 𝐴) < (𝐵 +_e 𝐴)))
58	49, 57	mpbid 147	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → ((𝑥 +_e -_e𝐴) +_e 𝐴) < (𝐵 +_e 𝐴))
59		xnpcan 10292	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝑥 +_e -_e𝐴) +_e 𝐴) = 𝑥)
60	50, 54, 59	syl2anc 415	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) → ((𝑥 +_e -_e𝐴) +_e 𝐴) = 𝑥)
61	60	adantr 276	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → ((𝑥 +_e -_e𝐴) +_e 𝐴) = 𝑥)
62		xaddcom 10281	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 +_e 𝐴) = (𝐴 +_e 𝐵))
63	45, 43, 62	syl2anc 415	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → (𝐵 +_e 𝐴) = (𝐴 +_e 𝐵))
64	58, 61, 63	3brtr3d 4162	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → 𝑥 < (𝐴 +_e 𝐵))
65		breq2 4135	. . . . 5 ⊢ (𝑦 = (𝐴 +_e 𝐵) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_e 𝐵)))
66	65	rspcev 2929	. . . 4 ⊢ (((𝐴 +_e 𝐵) ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)} ∧ 𝑥 < (𝐴 +_e 𝐵)) → ∃𝑦 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}𝑥 < 𝑦)
67	48, 64, 66	syl2anc 415	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐵) → ∃𝑦 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}𝑥 < 𝑦)
68	54	adantr 276	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → 𝐴 ∈ ℝ)
69	68, 3	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → 𝐴 ∈ ℝ^)
70		simpl3 1033	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → 𝐶 ∈ ℝ^)
71	70	adantr 276	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → 𝐶 ∈ ℝ^)
72	69, 71	xaddcld 10304	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → (𝐴 +_e 𝐶) ∈ ℝ^)
73		prid2g 3816	. . . . 5 ⊢ ((𝐴 +_e 𝐶) ∈ ℝ^* → (𝐴 +_e 𝐶) ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)})
74	72, 73	syl 14	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → (𝐴 +_e 𝐶) ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)})
75		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → (𝑥 +_e -_e𝐴) < 𝐶)
76	52	adantr 276	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → (𝑥 +_e -_e𝐴) ∈ ℝ^)
77		xltadd1 10296	. . . . . . 7 ⊢ (((𝑥 +_e -_e𝐴) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝑥 +_e -_e𝐴) < 𝐶 ↔ ((𝑥 +_e -_e𝐴) +_e 𝐴) < (𝐶 +_e 𝐴)))
78	76, 71, 68, 77	syl3anc 1278	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → ((𝑥 +_e -_e𝐴) < 𝐶 ↔ ((𝑥 +_e -_e𝐴) +_e 𝐴) < (𝐶 +_e 𝐴)))
79	75, 78	mpbid 147	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → ((𝑥 +_e -_e𝐴) +_e 𝐴) < (𝐶 +_e 𝐴))
80	60	adantr 276	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → ((𝑥 +_e -_e𝐴) +_e 𝐴) = 𝑥)
81		xaddcom 10281	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 +_e 𝐴) = (𝐴 +_e 𝐶))
82	71, 69, 81	syl2anc 415	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → (𝐶 +_e 𝐴) = (𝐴 +_e 𝐶))
83	79, 80, 82	3brtr3d 4162	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → 𝑥 < (𝐴 +_e 𝐶))
84		breq2 4135	. . . . 5 ⊢ (𝑦 = (𝐴 +_e 𝐶) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_e 𝐶)))
85	84	rspcev 2929	. . . 4 ⊢ (((𝐴 +_e 𝐶) ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)} ∧ 𝑥 < (𝐴 +_e 𝐶)) → ∃𝑦 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}𝑥 < 𝑦)
86	74, 83, 85	syl2anc 415	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_e -_e𝐴) < 𝐶) → ∃𝑦 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}𝑥 < 𝑦)
87		simprr 537	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))
88	10	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
89		rexneg 10250	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_e𝐴 = -𝐴)
90	89	3ad2ant1 1049	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_e𝐴 = -𝐴)
91	90	adantr 276	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) → -_e𝐴 = -𝐴)
92	54	renegcld 8715	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) → -𝐴 ∈ ℝ)
93	91, 92	eqeltrd 2315	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) → -_e𝐴 ∈ ℝ)
94		xltadd1 10296	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^ ∧ -_e𝐴 ∈ ℝ) → (𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ↔ (𝑥 +_e -_e𝐴) < ((𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) +_e -_e𝐴)))
95	50, 88, 93, 94	syl3anc 1278	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → (𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) ↔ (𝑥 +_e -_e𝐴) < ((𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )) +_e -_e𝐴)))
96	87, 95	mpbid 147	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → (𝑥 +_e -_e𝐴) < ((𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) +_e -_e𝐴))
97	3, 8	syl3an1 1311	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
98	97	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
99		xaddcom 10281	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^) → (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) = (sup({𝐵, 𝐶}, ℝ^, < ) +_e 𝐴))
100	42, 98, 99	syl2anc 415	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) = (sup({𝐵, 𝐶}, ℝ^*, < ) +_e 𝐴))
101	100	oveq1d 6103	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → ((𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )) +_e -_e𝐴) = ((sup({𝐵, 𝐶}, ℝ^*, < ) +_e 𝐴) +_e -_e𝐴))
102	96, 101	breqtrd 4157	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → (𝑥 +_e -_e𝐴) < ((sup({𝐵, 𝐶}, ℝ^, < ) +_e 𝐴) +_e -_e𝐴))
103		xpncan 10291	. . . . . 6 ⊢ ((sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^ ∧ 𝐴 ∈ ℝ) → ((sup({𝐵, 𝐶}, ℝ^, < ) +_e 𝐴) +_e -_e𝐴) = sup({𝐵, 𝐶}, ℝ^, < ))
104	98, 54, 103	syl2anc 415	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → ((sup({𝐵, 𝐶}, ℝ^, < ) +_e 𝐴) +_e -_e𝐴) = sup({𝐵, 𝐶}, ℝ^*, < ))
105	102, 104	breqtrd 4157	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → (𝑥 +_e -_e𝐴) < sup({𝐵, 𝐶}, ℝ^, < ))
106		xrltmaxsup 12049	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ (𝑥 +_e -_e𝐴) ∈ ℝ^) → ((𝑥 +_e -_e𝐴) < sup({𝐵, 𝐶}, ℝ^, < ) ↔ ((𝑥 +_e -_e𝐴) < 𝐵 ∨ (𝑥 +_e -_e𝐴) < 𝐶)))
107	44, 70, 52, 106	syl3anc 1278	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^, < )))) → ((𝑥 +_e -_e𝐴) < sup({𝐵, 𝐶}, ℝ^, < ) ↔ ((𝑥 +_e -_e𝐴) < 𝐵 ∨ (𝑥 +_e -_e𝐴) < 𝐶)))
108	105, 107	mpbid 147	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) → ((𝑥 +_e -_e𝐴) < 𝐵 ∨ (𝑥 +_e -_e𝐴) < 𝐶))
109	67, 86, 108	mpjaodan 810	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))) → ∃𝑦 ∈ {(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}𝑥 < 𝑦)
110	2, 10, 40, 109	eqsuptid 7344	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({(𝐴 +_e 𝐵), (𝐴 +_e 𝐶)}, ℝ^, < ) = (𝐴 +_e sup({𝐵, 𝐶}, ℝ^*, < )))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∃wrex 2529 {cpr 3710 class class class wbr 4131 (class class class)co 6088 supcsup 7329 ℝcr 8185 ℝ^*cxr 8366 < clt 8367 ≤ cle 8368 -cneg 8506 -_ecxne 10189 +_e cxad 10190

This proof depends on 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-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306

This proof 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-eu 2089 df-mo 2090 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-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-sup 7331 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-rp 10073 df-xneg 10192 df-xadd 10193 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788

This theorem is used by: xrmaxadd 12053

W3C validator