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Theorem xrmaxaddlem 11973

Description: Lemma for xrmaxadd 11974. 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 10152	. . 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 8335	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
4		simp1 1024	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
5		simp2 1025	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
6		simp3 1026	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7		xrmaxcl 11965	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
9	4, 8	xaddcld 10239	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
10	3, 9	syl3an1 1307	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
11		elpri 3717	. . . . 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 11966	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
14	5, 6, 13	syl2anc 411	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
15		xleadd2a 10229	. . . . . . . . 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 1277	. . . . . . . 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 4136	. . . . . 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 11967	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
21	5, 6, 20	syl2anc 411	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
22		xleadd2a 10229	. . . . . . . . 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 1277	. . . . . . . 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 4136	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
26	18, 25	jaodan 805	. . . . 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 10239	. . . . . . . . 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 2311	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → 𝑥 ∈ ℝ^)
31	4, 6	xaddcld 10239	. . . . . . . . 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 2311	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ∈ ℝ^)
34	30, 33	jaodan 805	. . . . . 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 8354	. . . . 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 411	. . . 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 1322	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}) → ¬ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) < 𝑥)
41	3	3ad2ant1 1045	. . . . . . . 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 1028	. . . . . . 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 10239	. . . . 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 3800	. . . . 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 531	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → 𝑥 ∈ ℝ^)
51	42	xnegcld 10210	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → -_𝑒𝐴 ∈ ℝ^)
52	50, 51	xaddcld 10239	. . . . . . . 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 1027	. . . . . . . 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 10231	. . . . . . 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 1274	. . . . . 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 10227	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝑥 +_𝑒 -_𝑒𝐴) +_𝑒 𝐴) = 𝑥)
60	50, 54, 59	syl2anc 411	. . . . . 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 10216	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐴 +_𝑒 𝐵))
63	45, 43, 62	syl2anc 411	. . . . 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 4145	. . . 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 4118	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐵) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐵)))
66	65	rspcev 2923	. . . 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 411	. . 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 1029	. . . . . . 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 10239	. . . . 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 3801	. . . . 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 10231	. . . . . . 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 1274	. . . . . 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 10216	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 +_𝑒 𝐴) = (𝐴 +_𝑒 𝐶))
82	71, 69, 81	syl2anc 411	. . . . 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 4145	. . . 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 4118	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐶) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐶)))
85	84	rspcev 2923	. . . 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 411	. . 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 533	. . . . . . 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 10185	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
90	89	3ad2ant1 1045	. . . . . . . . . 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 8671	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -𝐴 ∈ ℝ)
93	91, 92	eqeltrd 2311	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -_𝑒𝐴 ∈ ℝ)
94		xltadd1 10231	. . . . . . . 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 1274	. . . . . . 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 1307	. . . . . . . . 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 10216	. . . . . . . 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 411	. . . . . . 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 6073	. . . . . 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 4140	. . . . 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 10226	. . . . . 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 411	. . . . 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 4140	. . . 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 11970	. . . . 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 1274	. . . 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 806	. 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 7301	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

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ∃wrex 2523 {cpr 3695 class class class wbr 4114 (class class class)co 6058 supcsup 7286 ℝcr 8142 ℝ^*cxr 8323 < clt 8324 ≤ cle 8325 -cneg 8462 -_𝑒cxne 10124 +_𝑒 cxad 10125

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-rp 10008 df-xneg 10127 df-xadd 10128 df-seqfrec 10837 df-exp 10928 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712

This theorem is referenced by: xrmaxadd 11974

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