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Theorem xrmaxaddlem 11442

Description: Lemma for xrmaxadd 11443. 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 9889	. . 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 8089	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
4		simp1 999	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
5		simp2 1000	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
6		simp3 1001	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7		xrmaxcl 11434	. . . . 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 9976	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
10	3, 9	syl3an1 1282	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) ∈ ℝ^*)
11		elpri 3646	. . . . 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 11435	. . . . . . . . . 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 9966	. . . . . . . . 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 1252	. . . . . . . 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 4056	. . . . . 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 11436	. . . . . . . . . 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 9966	. . . . . . . . 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 1252	. . . . . . . 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 4056	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ≤ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))
26	18, 25	jaodan 798	. . . . 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 9976	. . . . . . . . 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 2273	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐵)) → 𝑥 ∈ ℝ^)
31	4, 6	xaddcld 9976	. . . . . . . . 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 2273	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 = (𝐴 +_𝑒 𝐶)) → 𝑥 ∈ ℝ^)
34	30, 33	jaodan 798	. . . . . 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 8108	. . . . 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 1297	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑥 ∈ {(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}) → ¬ (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )) < 𝑥)
41	3	3ad2ant1 1020	. . . . . . . 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 1003	. . . . . . 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 9976	. . . . 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 3727	. . . . 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 9947	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^, < )))) → -_𝑒𝐴 ∈ ℝ^)
52	50, 51	xaddcld 9976	. . . . . . . 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 1002	. . . . . . . 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 9968	. . . . . . 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 1249	. . . . . 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 9964	. . . . . . 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 9953	. . . . . 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 4065	. . . 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 4038	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐵) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐵)))
66	65	rspcev 2868	. . . 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 1004	. . . . . . 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 9976	. . . . 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 3728	. . . . 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 9968	. . . . . . 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 1249	. . . . . 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 9953	. . . . . 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 4065	. . . 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 4038	. . . . 5 ⊢ (𝑦 = (𝐴 +_𝑒 𝐶) → (𝑥 < 𝑦 ↔ 𝑥 < (𝐴 +_𝑒 𝐶)))
85	84	rspcev 2868	. . . 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 9922	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
90	89	3ad2ant1 1020	. . . . . . . . . 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 8423	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -𝐴 ∈ ℝ)
93	91, 92	eqeltrd 2273	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ ℝ^ ∧ 𝑥 < (𝐴 +_𝑒 sup({𝐵, 𝐶}, ℝ^*, < )))) → -_𝑒𝐴 ∈ ℝ)
94		xltadd1 9968	. . . . . . . 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 1249	. . . . . . 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 1282	. . . . . . . . 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 9953	. . . . . . . 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 5940	. . . . . 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 4060	. . . . 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 9963	. . . . . 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 4060	. . . 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 11439	. . . . 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 1249	. . . 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 799	. 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 7072	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 709 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 ∃wrex 2476 {cpr 3624 class class class wbr 4034 (class class class)co 5925 supcsup 7057 ℝcr 7895 ℝ^*cxr 8077 < clt 8078 ≤ cle 8079 -cneg 8215 -_𝑒cxne 9861 +_𝑒 cxad 9862

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-sup 7059 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-rp 9746 df-xneg 9864 df-xadd 9865 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181

This theorem is referenced by: xrmaxadd 11443

W3C validator