xrofsup - Mathbox for Thierry Arnoux

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Theorem xrofsup 30518

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 3914	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋 → 𝑥 ∈ ℝ^*))
4		xrofsup.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ ℝ^*)
5	4	sseld 3914	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝑌 → 𝑦 ∈ ℝ^*))
6	3, 5	anim12d 611	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)))
7	6	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*))
8		xaddcl 12620	. . . . . . 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 3156	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) ∈ ℝ^*)
11		fveq2 6645	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ( +_𝑒 ‘𝑢) = ( +_𝑒 ‘⟨𝑥, 𝑦⟩))
12		df-ov 7138	. . . . . . . 8 ⊢ (𝑥 +_𝑒 𝑦) = ( +_𝑒 ‘⟨𝑥, 𝑦⟩)
13	11, 12	eqtr4di 2851	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ( +_𝑒 ‘𝑢) = (𝑥 +_𝑒 𝑦))
14	13	eleq1d 2874	. . . . . 6 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (( +_𝑒 ‘𝑢) ∈ ℝ^* ↔ (𝑥 +_𝑒 𝑦) ∈ ℝ^*))
15	14	ralxp 5676	. . . . 5 ⊢ (∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) ∈ ℝ^*)
16	10, 15	sylibr 237	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^*)
17		xaddf 12605	. . . . . 6 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^
18		ffun 6490	. . . . . 6 ⊢ ( +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^ → Fun +_𝑒 )
19	17, 18	ax-mp 5	. . . . 5 ⊢ Fun +_𝑒
20		xpss12 5534	. . . . . . 7 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) → (𝑋 × 𝑌) ⊆ (ℝ^ × ℝ^*))
21	2, 4, 20	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (ℝ^* × ℝ^*))
22	17	fdmi 6498	. . . . . 6 ⊢ dom +_𝑒 = (ℝ^* × ℝ^*)
23	21, 22	sseqtrrdi 3966	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ dom +_𝑒 )
24		funimass4 6705	. . . . 5 ⊢ ((Fun +_𝑒 ∧ (𝑋 × 𝑌) ⊆ dom +_𝑒 ) → (( +_𝑒 “ (𝑋 × 𝑌)) ⊆ ℝ^* ↔ ∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^*))
25	19, 23, 24	sylancr 590	. . . 4 ⊢ (𝜑 → (( +_𝑒 “ (𝑋 × 𝑌)) ⊆ ℝ^* ↔ ∀𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) ∈ ℝ^*))
26	16, 25	mpbird 260	. . 3 ⊢ (𝜑 → ( +_𝑒 “ (𝑋 × 𝑌)) ⊆ ℝ^*)
27	1, 26	eqsstrd 3953	. 2 ⊢ (𝜑 → 𝑍 ⊆ ℝ^*)
28		supxrcl 12696	. . . 4 ⊢ (𝑋 ⊆ ℝ^* → sup(𝑋, ℝ^, < ) ∈ ℝ^)
29	2, 28	syl 17	. . 3 ⊢ (𝜑 → sup(𝑋, ℝ^, < ) ∈ ℝ^)
30		supxrcl 12696	. . . 4 ⊢ (𝑌 ⊆ ℝ^* → sup(𝑌, ℝ^, < ) ∈ ℝ^)
31	4, 30	syl 17	. . 3 ⊢ (𝜑 → sup(𝑌, ℝ^, < ) ∈ ℝ^)
32	29, 31	xaddcld 12682	. 2 ⊢ (𝜑 → (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) ∈ ℝ^*)
33	1	eleq2d 2875	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
34	33	pm5.32i 578	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) ↔ (𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
35		nfvd 1916	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → Ⅎ𝑥 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
36		nfvd 1916	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → Ⅎ𝑦 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
37	2	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑋 ⊆ ℝ^*)
38		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋)
39		supxrub 12705	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑥 ∈ 𝑋) → 𝑥 ≤ sup(𝑋, ℝ^*, < ))
40	37, 38, 39	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ≤ sup(𝑋, ℝ^*, < ))
41	4	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑌 ⊆ ℝ^*)
42		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
43		supxrub 12705	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ ℝ^* ∧ 𝑦 ∈ 𝑌) → 𝑦 ≤ sup(𝑌, ℝ^*, < ))
44	41, 42, 43	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ≤ sup(𝑌, ℝ^*, < ))
45	37, 38	sseldd 3916	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ ℝ^*)
46	41, 42	sseldd 3916	. . . . . . . . . . 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 12640	. . . . . . . . . . 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 837	. . . . . . . . . 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 698	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
52	51	ralrimivva 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
53		fvelima 6706	. . . . . . . . . . 11 ⊢ ((Fun +_𝑒 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧)
54	19, 53	mpan 689	. . . . . . . . . 10 ⊢ (𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌)) → ∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧)
55	54	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧)
56	13	eqeq1d 2800	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (( +_𝑒 ‘𝑢) = 𝑧 ↔ (𝑥 +_𝑒 𝑦) = 𝑧))
57	56	rexxp 5677	. . . . . . . . 9 ⊢ (∃𝑢 ∈ (𝑋 × 𝑌)( +_𝑒 ‘𝑢) = 𝑧 ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) = 𝑧)
58	55, 57	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 (𝑥 +_𝑒 𝑦) = 𝑧)
59	52, 58	r19.29d2r 3291	. . . . . . 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 464	. . . . . . . 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 3211	. . . . . . 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 221	. . . . . 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 5033	. . . . . . . . 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 480	. . . . . . . 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 3206	. . . . . . 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 3206	. . . . . 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 30243	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( +_𝑒 “ (𝑋 × 𝑌))) → 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
69	34, 68	sylbi 220	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
70	69	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝑧 ≤ (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )))
71	2	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑋 ⊆ ℝ^*)
72	4	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑌 ⊆ ℝ^*)
73		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → 𝑧 ∈ ℝ)
74	29	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → sup(𝑋, ℝ^, < ) ∈ ℝ^)
75	31	ad2antrr 725	. . . . . . 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 725	. . . . . . 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 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → sup(𝑌, ℝ^*, < ) ≠ -∞)
80		simpr 488	. . . . . . 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 30508	. . . . . 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑏(𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^*)
83		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑏ℝ^*
84		nfre1 3265	. . . . . . . . 9 ⊢ Ⅎ𝑏∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))
85	83, 84	nfrex 3268	. . . . . . . 8 ⊢ Ⅎ𝑏∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < ))
86	82, 85	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑏((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) ∧ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* (𝑧 = (𝑎 +_𝑒 𝑏) ∧ 𝑎 < sup(𝑋, ℝ^, < ) ∧ 𝑏 < sup(𝑌, ℝ^, < )))
87		nfvd 1916	. . . . . . 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 1916	. . . . . . 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 3150	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) → ∀𝑏 ∈ ℝ^ (𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^*))
91	90	ralrimivw 3150	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^) → ∀𝑎 ∈ ℝ^ ∀𝑏 ∈ ℝ^* (𝑋 ⊆ ℝ^* ∧ 𝑌 ⊆ ℝ^*))
92	91	adantr 484	. . . . . . . . 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 488	. . . . . . . . 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 3291	. . . . . . . 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 777	. . . . . . . . . . . . . . 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 1357	. . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . 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 1357	. . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . 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 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)) → 𝑤 ∈ 𝑌)
106	104, 105	sseldd 3916	. . . . . . . . . . . . . 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 519	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 12641	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑤 ∈ ℝ^*)) → ((𝑎 < 𝑣 ∧ 𝑏 < 𝑤) → (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤)))
110	109	imp 410	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑤 ∈ ℝ^*)) ∧ (𝑎 < 𝑣 ∧ 𝑏 < 𝑤)) → (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤))
111		breq1 5033	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → (𝑧 < (𝑣 +_𝑒 𝑤) ↔ (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤)))
112	111	biimpar 481	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = (𝑎 +_𝑒 𝑏) ∧ (𝑎 +_𝑒 𝑏) < (𝑣 +_𝑒 𝑤)) → 𝑧 < (𝑣 +_𝑒 𝑤))
113	110, 112	sylan2 595	. . . . . . . . . . . . 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 835	. . . . . . . . . . . 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 774	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . 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 12706	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ⊆ ℝ^* ∧ 𝑎 ∈ ℝ^) → (𝑎 < sup(𝑋, ℝ^, < ) ↔ ∃𝑣 ∈ 𝑋 𝑎 < 𝑣))
119	118	biimpa 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ⊆ ℝ^* ∧ 𝑎 ∈ ℝ^) ∧ 𝑎 < sup(𝑋, ℝ^, < )) → ∃𝑣 ∈ 𝑋 𝑎 < 𝑣)
120	115, 116, 117, 119	syl21anc 836	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . 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 12706	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ⊆ ℝ^* ∧ 𝑏 ∈ ℝ^) → (𝑏 < sup(𝑌, ℝ^, < ) ↔ ∃𝑤 ∈ 𝑌 𝑏 < 𝑤))
125	124	biimpa 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑌 ⊆ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑏 < sup(𝑌, ℝ^, < )) → ∃𝑤 ∈ 𝑌 𝑏 < 𝑤)
126	121, 122, 123, 125	syl21anc 836	. . . . . . . . . . . . . 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 3320	. . . . . . . . . . . . . 14 ⊢ (∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 (𝑎 < 𝑣 ∧ 𝑏 < 𝑤) ↔ (∃𝑣 ∈ 𝑋 𝑎 < 𝑣 ∧ ∃𝑤 ∈ 𝑌 𝑏 < 𝑤))
128	120, 126, 127	sylanbrc 586	. . . . . . . . . . . . 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 462	. . . . . . . . . . . 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 3237	. . . . . . . . . . 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 416	. . . . . . . . . 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 3233	. . . . . . . . 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 3205	. . . . . . . 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 30242	. . . . . 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 836	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < ))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤))
137		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑋)
138		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑌)
139	19	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → Fun +_𝑒 )
140	23	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑋 × 𝑌) ⊆ dom +_𝑒 )
141	137, 138, 139, 140	elovimad 7183	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑣 +_𝑒 𝑤) ∈ ( +_𝑒 “ (𝑋 × 𝑌)))
142	1	eleq2d 2875	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑣 +_𝑒 𝑤) ∈ 𝑍 ↔ (𝑣 +_𝑒 𝑤) ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
143	142	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 +_𝑒 𝑤) ∈ 𝑍 ↔ (𝑣 +_𝑒 𝑤) ∈ ( +_𝑒 “ (𝑋 × 𝑌))))
144	141, 143	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑣 +_𝑒 𝑤) ∈ 𝑍)
145		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) ∧ 𝑘 = (𝑣 +_𝑒 𝑤)) → 𝑘 = (𝑣 +_𝑒 𝑤))
146	145	breq2d 5042	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) ∧ 𝑘 = (𝑣 +_𝑒 𝑤)) → (𝑧 < 𝑘 ↔ 𝑧 < (𝑣 +_𝑒 𝑤)))
147	144, 146	rspcedv 3564	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) → (𝑧 < (𝑣 +_𝑒 𝑤) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
148	147	rexlimdvva 3253	. . . . . 6 ⊢ (𝜑 → (∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑌 𝑧 < (𝑣 +_𝑒 𝑤) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
149	148	ad2antrr 725	. . . . 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 416	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
152	151	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ (𝑧 < (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^, < )) → ∃𝑘 ∈ 𝑍 𝑧 < 𝑘))
153		supxr2 12695	. 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 837	1 ⊢ (𝜑 → sup(𝑍, ℝ^, < ) = (sup(𝑋, ℝ^, < ) +_𝑒 sup(𝑌, ℝ^*, < )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ⟨cop 4531 class class class wbr 5030 × cxp 5517 dom cdm 5519 “ cima 5522 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℝcr 10525 -∞cmnf 10662 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 +_𝑒 cxad 12493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-rp 12378 df-xneg 12495 df-xadd 12496

This theorem is referenced by: (None)

W3C validator