xrofsup - Mathbox for Thierry Arnoux

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Theorem xrofsup 30098

Description: The supremum is preserved by extended addition set operation. (Provided minus infinity is not involved as it does not behave well with addition.) (Contributed by Thierry Arnoux, 20-Mar-2017.)

**Hypotheses**
Ref	Expression
xrofsup.1	⊢ (𝜑 → 𝑋 ⊆ ℝ^*)
xrofsup.2	⊢ (𝜑 → 𝑌 ⊆ ℝ^*)
xrofsup.3	⊢ (𝜑 → sup(𝑋, ℝ^*, < ) ≠ -∞)
xrofsup.4	⊢ (𝜑 → sup(𝑌, ℝ^*, < ) ≠ -∞)
xrofsup.5	⊢ (𝜑 → 𝑍 = ( +_𝑒 “ (𝑋 × 𝑌)))

**Assertion**
Ref	Expression
xrofsup	⊢ (𝜑 → sup(𝑍, ℝ^, < ) = (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^*, < )))

Proof of Theorem xrofsup Dummy variables 𝑎 𝑏 𝑘 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrofsup.5	. . 3 ⊢ (𝜑 → 𝑍 = ( +_𝑒 “ (𝑋 × 𝑌)))
2		xrofsup.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℝ^*)
3	2	sseld 3819	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋 → 𝑥 ∈ ℝ^*))
4		xrofsup.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ ℝ^*)
5	4	sseld 3819	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝑌 → 𝑦 ∈ ℝ^*))
6	3, 5	anim12d 602	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)))
7	6	imp 397	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*))
8		xaddcl 12382	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥 +_𝑒 𝑦) ∈ ℝ^)
9	7, 8	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥 +_𝑒 𝑦) ∈ ℝ^*)
10	9	ralrimivva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) ∈ ℝ^*)
11		fveq2 6446	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ( +_𝑒 ‘𝑢) = ( +_𝑒 ‘⟨𝑥, 𝑦⟩))
12		df-ov 6925	. . . . . . . 8 ⊢ (𝑥 +_𝑒 𝑦) = ( +_𝑒 ‘⟨𝑥, 𝑦⟩)
13	11, 12	syl6eqr 2831	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ( +_𝑒 ‘𝑢) = (𝑥 +_𝑒 𝑦))
14	13	eleq1d 2843	. . . . . 6 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (( +_𝑒 ‘𝑢) ∈ ℝ^* ↔ (𝑥 +_𝑒 𝑦) ∈ ℝ^*))
15	14	ralxp 5509	. . . . 5 ⊢ (∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) ∈ ℝ^*)
16	10, 15	sylibr 226	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^*)
17		xaddf 12367	. . . . . 6 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^
18		ffun 6294	. . . . . 6 ⊢ ( +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^ → Fun +_𝑒 )
19	17, 18	ax-mp 5	. . . . 5 ⊢ Fun +_𝑒
20		xpss12 5370	. . . . . . 7 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) → (𝑋 × 𝑌) ⊆ (ℝ^ × ℝ^*))
21	2, 4, 20	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (ℝ^* × ℝ^*))
22	17	fdmi 6301	. . . . . 6 ⊢ dom +_𝑒 = (ℝ^* × ℝ^*)
23	21, 22	syl6sseqr 3870	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ dom +_𝑒 )
24		funimass4 6507	. . . . 5 ⊢ ((Fun +_𝑒 ∧ (𝑋 × 𝑌) ⊆ dom +_𝑒 ) → (( +_𝑒 “ (𝑋 × 𝑌)) ⊆ ℝ^* ↔ ∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^*))
25	19, 23, 24	sylancr 581	. . . 4 ⊢ (𝜑 → (( +_𝑒 “ (𝑋 × 𝑌)) ⊆ ℝ^* ↔ ∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^*))
26	16, 25	mpbird 249	. . 3 ⊢ (𝜑 → ( +_𝑒 “ (𝑋 × 𝑌)) ⊆ ℝ^*)
27	1, 26	eqsstrd 3857	. 2 ⊢ (𝜑 → 𝑍 ⊆ ℝ^*)
28		supxrcl 12457	. . . 4 ⊢ (𝑋 ⊆ ℝ^* → sup(𝑋, ℝ^, < ) ∈ ℝ^)
29	2, 28	syl 17	. . 3 ⊢ (𝜑 → sup(𝑋, ℝ^, < ) ∈ ℝ^)
30		supxrcl 12457	. . . 4 ⊢ (𝑌 ⊆ ℝ^* → sup(𝑌, ℝ^, < ) ∈ ℝ^)
31	4, 30	syl 17	. . 3 ⊢ (𝜑 → sup(𝑌, ℝ^, < ) ∈ ℝ^)
32	29, 31	xaddcld 12443	. 2 ⊢ (𝜑 → (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ∈ ℝ^*)
33	1	eleq2d 2844	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
34	33	pm5.32i 570	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) ↔ (𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
35		nfvd 1958	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → Ⅎ𝑥 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
36		nfvd 1958	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → Ⅎ𝑦 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
37	2	ad2antrr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑋 ⊆ ℝ^*)
38		simprl 761	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋)
39		supxrub 12466	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑥 ∈ 𝑋) → 𝑥 ≤ sup(𝑋, ℝ^*, < ))
40	37, 38, 39	syl2anc 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ≤ sup(𝑋, ℝ^*, < ))
41	4	ad2antrr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑌 ⊆ ℝ^*)
42		simprr 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
43		supxrub 12466	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ ℝ^* ∧ 𝑦 ∈ 𝑌) → 𝑦 ≤ sup(𝑌, ℝ^*, < ))
44	41, 42, 43	syl2anc 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ≤ sup(𝑌, ℝ^*, < ))
45	37, 38	sseldd 3821	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ ℝ^*)
46	41, 42	sseldd 3821	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ ℝ^*)
47	37, 28	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → sup(𝑋, ℝ^, < ) ∈ ℝ^)
48	41, 30	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → sup(𝑌, ℝ^, < ) ∈ ℝ^)
49		xle2add 12401	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) ∧ (sup(𝑋, ℝ^, < ) ∈ ℝ^* ∧ sup(𝑌, ℝ^, < ) ∈ ℝ^)) → ((𝑥 ≤ sup(𝑋, ℝ^, < ) ∧ 𝑦 ≤ sup(𝑌, ℝ^, < )) → (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))))
50	45, 46, 47, 48, 49	syl22anc 829	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ≤ sup(𝑋, ℝ^, < ) ∧ 𝑦 ≤ sup(𝑌, ℝ^, < )) → (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))))
51	40, 44, 50	mp2and 689	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
52	51	ralrimivva 3152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
53		fvelima 6508	. . . . . . . . . . 11 ⊢ ((Fun +_𝑒 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧)
54	19, 53	mpan 680	. . . . . . . . . 10 ⊢ (𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌)) → ∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧)
55	54	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧)
56	13	eqeq1d 2779	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (( +_𝑒 ‘𝑢) = 𝑧 ↔ (𝑥 +_𝑒 𝑦) = 𝑧))
57	56	rexxp 5510	. . . . . . . . 9 ⊢ (∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧 ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) = 𝑧)
58	55, 57	sylib 210	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) = 𝑧)
59	52, 58	r19.29d2r 3265	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ((𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ∧ (𝑥 +_𝑒 𝑦) = 𝑧))
60		ancom 454	. . . . . . . 8 ⊢ (((𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ∧ (𝑥 +_𝑒 𝑦) = 𝑧) ↔ ((𝑥 +_𝑒 𝑦) = 𝑧 ∧ (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))))
61	60	2rexbii 3224	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ((𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ∧ (𝑥 +_𝑒 𝑦) = 𝑧) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ((𝑥 +_𝑒 𝑦) = 𝑧 ∧ (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))))
62	59, 61	sylib 210	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ((𝑥 +_𝑒 𝑦) = 𝑧 ∧ (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))))
63		breq1 4889	. . . . . . . . 9 ⊢ ((𝑥 +_𝑒 𝑦) = 𝑧 → ((𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ↔ 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))))
64	63	biimpa 470	. . . . . . . 8 ⊢ (((𝑥 +_𝑒 𝑦) = 𝑧 ∧ (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
65	64	reximi 3191	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑌 ((𝑥 +_𝑒 𝑦) = 𝑧 ∧ (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → ∃𝑦 ∈ 𝑌 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
66	65	reximi 3191	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ((𝑥 +_𝑒 𝑦) = 𝑧 ∧ (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
67	62, 66	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
68	35, 36, 67	19.9d2r 29891	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
69	34, 68	sylbi 209	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
70	69	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
71	2	ad2antrr 716	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑋 ⊆ ℝ^*)
72	4	ad2antrr 716	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑌 ⊆ ℝ^*)
73		simplr 759	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑧 ∈ ℝ)
74	29	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → sup(𝑋, ℝ^, < ) ∈ ℝ^)
75	31	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → sup(𝑌, ℝ^, < ) ∈ ℝ^)
76		xrofsup.3	. . . . . . . 8 ⊢ (𝜑 → sup(𝑋, ℝ^*, < ) ≠ -∞)
77	76	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → sup(𝑋, ℝ^*, < ) ≠ -∞)
78		xrofsup.4	. . . . . . . 8 ⊢ (𝜑 → sup(𝑌, ℝ^*, < ) ≠ -∞)
79	78	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → sup(𝑌, ℝ^*, < ) ≠ -∞)
80		simpr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
81	73, 74, 75, 77, 79, 80	xlt2addrd 30088	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → ∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))
82		nfv 1957	. . . . . . . 8 ⊢ Ⅎ𝑏(𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^*)
83		nfcv 2933	. . . . . . . . 9 ⊢ Ⅎ𝑏ℝ^*
84		nfre1 3185	. . . . . . . . 9 ⊢ Ⅎ𝑏∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))
85	83, 84	nfrex 3187	. . . . . . . 8 ⊢ Ⅎ𝑏∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))
86	82, 85	nfan 1946	. . . . . . 7 ⊢ Ⅎ𝑏((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))
87		nfvd 1958	. . . . . . 7 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → Ⅎ𝑎∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
88		nfvd 1958	. . . . . . 7 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → Ⅎ𝑏∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
89		id 22	. . . . . . . . . . . 12 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) → (𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^*))
90	89	ralrimivw 3148	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) → ∀𝑏 ∈ ℝ^ (𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^*))
91	90	ralrimivw 3148	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) → ∀𝑎 ∈ ℝ^ ∀𝑏 ∈ ℝ^* (𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^*))
92	91	adantr 474	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → ∀𝑎 ∈ ℝ^* ∀𝑏 ∈ ℝ^* (𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^*))
93		simpr 479	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → ∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))
94	92, 93	r19.29d2r 3265	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → ∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < ))))
95		simplrr 768	. . . . . . . . . . . . . . 15 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌 ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤))) → (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))
96	95	3anassrs 1422	. . . . . . . . . . . . . 14 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))
97	96	simp1d 1133	. . . . . . . . . . . . 13 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑧 = (𝑎 +_𝑒 𝑏))
98		simp-4l 773	. . . . . . . . . . . . . 14 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^*))
99		simplrl 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌 ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤))) → (𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^*))
100	99	3anassrs 1422	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → (𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^*))
101	100	simpld 490	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑋 ⊆ ℝ^)
102		simpllr 766	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑣 ∈ 𝑋)
103	101, 102	sseldd 3821	. . . . . . . . . . . . . 14 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑣 ∈ ℝ^)
104	100	simprd 491	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑌 ⊆ ℝ^)
105		simplr 759	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑤 ∈ 𝑌)
106	104, 105	sseldd 3821	. . . . . . . . . . . . . 14 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑤 ∈ ℝ^)
107	98, 103, 106	jca32 511	. . . . . . . . . . . . 13 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → ((𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑤 ∈ ℝ^*)))
108		simpr 479	. . . . . . . . . . . . 13 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → (𝑎 < 𝑣 ∧ 𝑏 < 𝑤))
109		xlt2add 12402	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑤 ∈ ℝ^*)) → ((𝑎 < 𝑣 ∧ 𝑏 < 𝑤) → (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤)))
110	109	imp 397	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑤 ∈ ℝ^*)) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤))
111		breq1 4889	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → (𝑧 < (𝑣 +_𝑒 𝑤) ↔ (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤)))
112	111	biimpar 471	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = (𝑎 +_𝑒 𝑏) ∧ (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤)) → 𝑧 < (𝑣 +_𝑒 𝑤))
113	110, 112	sylan2 586	. . . . . . . . . . . . 13 ⊢ ((𝑧 = (𝑎 +_𝑒 𝑏) ∧ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑤 ∈ ℝ^*)) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤))) → 𝑧 < (𝑣 +_𝑒 𝑤))
114	97, 107, 108, 113	syl12anc 827	. . . . . . . . . . . 12 ⊢ ((((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))) ∧ 𝑣 ∈ 𝑋) ∧ 𝑤 ∈ 𝑌) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → 𝑧 < (𝑣 +_𝑒 𝑤))
115		simplll 765	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^)) → 𝑋 ⊆ ℝ^)
116		simprl 761	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^)) → 𝑎 ∈ ℝ^)
117		simplr2 1234	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^)) → 𝑎 < sup(𝑋, ℝ^, < ))
118		supxrlub 12467	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑎 ∈ ℝ^) → (𝑎 < sup(𝑋, ℝ^, < ) ↔ ∃𝑣 ∈ 𝑋 𝑎 < 𝑣))
119	118	biimpa 470	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑎 ∈ ℝ^) ∧ 𝑎 < sup(𝑋, ℝ^, < )) → ∃𝑣 ∈ 𝑋 𝑎 < 𝑣)
120	115, 116, 117, 119	syl21anc 828	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^*)) → ∃𝑣 ∈ 𝑋 𝑎 < 𝑣)
121		simpllr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^)) → 𝑌 ⊆ ℝ^)
122		simprr 763	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^)) → 𝑏 ∈ ℝ^)
123		simplr3 1236	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^)) → 𝑏 < sup(𝑌, ℝ^, < ))
124		supxrlub 12467	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ⊆ ℝ^* ∧ 𝑏 ∈ ℝ^) → (𝑏 < sup(𝑌, ℝ^, < ) ↔ ∃𝑤 ∈ 𝑌 𝑏 < 𝑤))
125	124	biimpa 470	. . . . . . . . . . . . . . 15 ⊢ (((𝑌 ⊆ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑏 < sup(𝑌, ℝ^, < )) → ∃𝑤 ∈ 𝑌 𝑏 < 𝑤)
126	121, 122, 123, 125	syl21anc 828	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^*)) → ∃𝑤 ∈ 𝑌 𝑏 < 𝑤)
127		reeanv 3292	. . . . . . . . . . . . . 14 ⊢ (∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 (𝑎 < 𝑣 ∧ 𝑏 < 𝑤) ↔ (∃𝑣 ∈ 𝑋 𝑎 < 𝑣 ∧ ∃𝑤 ∈ 𝑌 𝑏 < 𝑤))
128	120, 126, 127	sylanbrc 578	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) ∧ (𝑎 ∈ ℝ^ ∧ 𝑏 ∈ ℝ^*)) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 (𝑎 < 𝑣 ∧ 𝑏 < 𝑤))
129	128	ancoms 452	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 (𝑎 < 𝑣 ∧ 𝑏 < 𝑤))
130	114, 129	reximddv2 3201	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ ((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < )))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
131	130	ex 403	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → (((𝑋 ⊆ ℝ^ ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^*, < ))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤)))
132	131	reximdva 3197	. . . . . . . . 9 ⊢ (𝑎 ∈ ℝ^* → (∃𝑏 ∈ ℝ^* ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → ∃𝑏 ∈ ℝ^ ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤)))
133	132	reximia 3189	. . . . . . . 8 ⊢ (∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
134	94, 133	syl 17	. . . . . . 7 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → ∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
135	86, 87, 88, 134	19.9d2rf 29890	. . . . . 6 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
136	71, 72, 81, 135	syl21anc 828	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
137		simprl 761	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑋)
138		simprr 763	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑌)
139	19	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → Fun +_𝑒 )
140	23	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑋 × 𝑌) ⊆ dom +_𝑒 )
141	137, 138, 139, 140	elovimad 6969	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑣 +_𝑒 𝑤) ∈ ( +_𝑒 “ (𝑋 × 𝑌)))
142	1	eleq2d 2844	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑣 +_𝑒 𝑤) ∈ 𝑍 ↔ (𝑣 +_𝑒 𝑤) ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
143	142	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 +_𝑒 𝑤) ∈ 𝑍 ↔ (𝑣 +_𝑒 𝑤) ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
144	141, 143	mpbird 249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑣 +_𝑒 𝑤) ∈ 𝑍)
145		simpr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) ∧ 𝑘 = (𝑣 +_𝑒 𝑤)) → 𝑘 = (𝑣 +_𝑒 𝑤))
146	145	breq2d 4898	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) ∧ 𝑘 = (𝑣 +_𝑒 𝑤)) → (𝑧 < 𝑘 ↔ 𝑧 < (𝑣 +_𝑒 𝑤)))
147	144, 146	rspcedv 3514	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑧 < (𝑣 +_𝑒 𝑤) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
148	147	rexlimdvva 3220	. . . . . 6 ⊢ (𝜑 → (∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
149	148	ad2antrr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → (∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
150	136, 149	mpd 15	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘)
151	150	ex 403	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
152	151	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ (𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
153		supxr2 12456	. 2 ⊢ (((𝑍 ⊆ ℝ^* ∧ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ∈ ℝ^) ∧ (∀𝑧 ∈ 𝑍 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ∧ ∀𝑧 ∈ ℝ (𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))) → sup(𝑍, ℝ^, < ) = (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
154	27, 32, 70, 152, 153	syl22anc 829	1 ⊢ (𝜑 → sup(𝑍, ℝ^, < ) = (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^*, < )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 ⊆ wss 3791 ⟨cop 4403 class class class wbr 4886 × cxp 5353 dom cdm 5355 “ cima 5358 Fun wfun 6129 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 supcsup 8634 ℝcr 10271 -∞cmnf 10409 ℝ^*cxr 10410 < clt 10411 ≤ cle 10412 +_𝑒 cxad 12255

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-2 11438 df-rp 12138 df-xneg 12257 df-xadd 12258

This theorem is referenced by: (None)

W3C validator