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Theorem xrmaxaddlem 11201

Description: Lemma for xrmaxadd 11202. 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 9733	. . 3 ⊢ ((𝑓 ∈ ℝ^* ∧ 𝑔 ∈ ℝ^*) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2	1	adantl 275	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑓 ∈ ℝ^ ∧ 𝑔 ∈ ℝ^*)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3		rexr 7944	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
4		simp1 987	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
5		simp2 988	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
6		simp3 989	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7		xrmaxcl 11193	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
8	5, 6, 7	syl2anc 409	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
9	4, 8	xaddcld 9820	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
10	3, 9	syl3an1 1261	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
11		elpri 3599	. . . . 5 ⊢ (𝑥 ∈ {(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)} → (𝑥 = (𝐴 +_𝑒 𝐵) ∨ 𝑥 = (𝐴 +_𝑒 𝐶)))
12		simpr 109	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → 𝑥 = (𝐴 +_𝑒 𝐵))
13		xrmax1sup 11194	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
14	5, 6, 13	syl2anc 409	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
15		xleadd2a 9810	. . . . . . . . 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 1231	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
17	16	adantr 274	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → (𝐴 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
18	12, 17	eqbrtrd 4004	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → 𝑥 ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
19		simpr 109	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 = (𝐴 +_𝑒 𝐶))
20		xrmax2sup 11195	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
21	5, 6, 20	syl2anc 409	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ≤ sup({𝐵, 𝐶}, ℝ^, < ))
22		xleadd2a 9810	. . . . . . . . 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 1231	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
24	23	adantr 274	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → (𝐴 +_𝑒 𝐶) ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
25	19, 24	eqbrtrd 4004	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
26	18, 25	jaodan 787	. . . . 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 284	. . . 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 9820	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
29	28	adantr 274	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
30	12, 29	eqeltrd 2243	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → 𝑥 ∈ ℝ^)
31	4, 6	xaddcld 9820	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
32	31	adantr 274	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
33	19, 32	eqeltrd 2243	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ∈ ℝ^)
34	30, 33	jaodan 787	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 = (𝐴 +_𝑒 𝐵) ∨ 𝑥 = (𝐴 +_𝑒 𝐶))) → 𝑥 ∈ ℝ^)
35	11, 34	sylan2 284	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}) → 𝑥 ∈ ℝ^)
36	9	adantr 274	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
37		xrlenlt 7963	. . . . 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 409	. . . 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 146	. . 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 1276	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}) → ¬ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) < 𝑥)
41	3	3ad2ant1 1008	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
42	41	adantr 274	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → 𝐴 ∈ ℝ^)
43	42	adantr 274	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) ∧ (𝑥 +_𝑒 -_𝑒𝐴) < 𝐵) → 𝐴 ∈ ℝ^)
44		simpl2 991	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → 𝐵 ∈ ℝ^)
45	44	adantr 274	. . . . . 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 9820	. . . . 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 3680	. . . . 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 109	. . . . . 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 521	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → 𝑥 ∈ ℝ^)
51	42	xnegcld 9791	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → -_𝑒𝐴 ∈ ℝ^)
52	50, 51	xaddcld 9820	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → (𝑥 +_𝑒 -_𝑒𝐴) ∈ ℝ^)
53	52	adantr 274	. . . . . . 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 990	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → 𝐴 ∈ ℝ)
55	54	adantr 274	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) ∧ (𝑥 +_𝑒 -_𝑒𝐴) < 𝐵) → 𝐴 ∈ ℝ)
56		xltadd1 9812	. . . . . . 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 1228	. . . . . 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 146	. . . . 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 9808	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝑥 +_𝑒 -_𝑒𝐴) +_𝑒 𝐴) = 𝑥)
60	50, 54, 59	syl2anc 409	. . . . . 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 274	. . . . 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 9797	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐴 +_𝑒 𝐵))
63	45, 43, 62	syl2anc 409	. . . . 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 4013	. . . 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 3986	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐵) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐵)))
66	65	rspcev 2830	. . . 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 409	. . 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 274	. . . . . . 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 992	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → 𝐶 ∈ ℝ^)
71	70	adantr 274	. . . . . 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 9820	. . . . 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 3681	. . . . 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 109	. . . . . 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 274	. . . . . . 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 9812	. . . . . . 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 1228	. . . . . 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 146	. . . . 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 274	. . . . 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 9797	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 +_𝑒 𝐴) = (𝐴 +_𝑒 𝐶))
82	71, 69, 81	syl2anc 409	. . . . 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 4013	. . . 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 3986	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐶) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐶)))
85	84	rspcev 2830	. . . 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 409	. . 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 522	. . . . . . 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 274	. . . . . . . 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 9766	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
90	89	3ad2ant1 1008	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒𝐴 = -𝐴)
91	90	adantr 274	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -_𝑒𝐴 = -𝐴)
92	54	renegcld 8278	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -𝐴 ∈ ℝ)
93	91, 92	eqeltrd 2243	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -_𝑒𝐴 ∈ ℝ)
94		xltadd1 9812	. . . . . . . 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 1228	. . . . . . 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 146	. . . . . 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 1261	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
98	97	adantr 274	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → sup({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
99		xaddcom 9797	. . . . . . . 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 409	. . . . . . 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 5857	. . . . . 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 4008	. . . . 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 9807	. . . . . 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 409	. . . . 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 4008	. . . 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 11198	. . . . 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 1228	. . . 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 146	. . 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 788	. 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 6962	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 103 ↔ wb 104 ∨ wo 698 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ∃wrex 2445 {cpr 3577 class class class wbr 3982 (class class class)co 5842 supcsup 6947 ℝcr 7752 ℝ^*cxr 7932 < clt 7933 ≤ cle 7934 -cneg 8070 -_𝑒cxne 9705 +_𝑒 cxad 9706

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-sup 6949 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-rp 9590 df-xneg 9708 df-xadd 9709 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941

This theorem is referenced by: xrmaxadd 11202

W3C validator