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Theorem xrmaxaddlem 11268

Description: Lemma for xrmaxadd 11269. 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 9797	. . 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 8003	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
4		simp1 997	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
5		simp2 998	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
6		simp3 999	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7		xrmaxcl 11260	. . . . 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 9884	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
10	3, 9	syl3an1 1271	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
11		elpri 3616	. . . . 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 11261	. . . . . . . . . 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 9874	. . . . . . . . 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 1241	. . . . . . . 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 4026	. . . . . 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 11262	. . . . . . . . . 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 9874	. . . . . . . . 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 1241	. . . . . . . 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 4026	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
26	18, 25	jaodan 797	. . . . 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 9884	. . . . . . . . 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 2254	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → 𝑥 ∈ ℝ^)
31	4, 6	xaddcld 9884	. . . . . . . . 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 2254	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ∈ ℝ^)
34	30, 33	jaodan 797	. . . . . 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 8022	. . . . 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 1286	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}) → ¬ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) < 𝑥)
41	3	3ad2ant1 1018	. . . . . . . 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 1001	. . . . . . 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 9884	. . . . 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 3697	. . . . 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 529	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → 𝑥 ∈ ℝ^)
51	42	xnegcld 9855	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → -_𝑒𝐴 ∈ ℝ^)
52	50, 51	xaddcld 9884	. . . . . . . 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 1000	. . . . . . . 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 9876	. . . . . . 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 1238	. . . . . 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 9872	. . . . . . 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 9861	. . . . . 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 4035	. . . 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 4008	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐵) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐵)))
66	65	rspcev 2842	. . . 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 1002	. . . . . . 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 9884	. . . . 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 3698	. . . . 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 9876	. . . . . . 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 1238	. . . . . 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 9861	. . . . . 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 4035	. . . 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 4008	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐶) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐶)))
85	84	rspcev 2842	. . . 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 531	. . . . . . 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 9830	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
90	89	3ad2ant1 1018	. . . . . . . . . 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 8337	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -𝐴 ∈ ℝ)
93	91, 92	eqeltrd 2254	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -_𝑒𝐴 ∈ ℝ)
94		xltadd1 9876	. . . . . . . 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 1238	. . . . . . 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 1271	. . . . . . . . 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 9861	. . . . . . . 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 5890	. . . . . 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 4030	. . . . 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 9871	. . . . . 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 4030	. . . 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 11265	. . . . 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 1238	. . . 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 798	. 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 6996	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 708 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 {cpr 3594 class class class wbr 4004 (class class class)co 5875 supcsup 6981 ℝcr 7810 ℝ^*cxr 7991 < clt 7992 ≤ cle 7993 -cneg 8129 -_𝑒cxne 9769 +_𝑒 cxad 9770

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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-sup 6983 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-rp 9654 df-xneg 9772 df-xadd 9773 df-seqfrec 10446 df-exp 10520 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008

This theorem is referenced by: xrmaxadd 11269

W3C validator