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Theorem genprndl 7882

Description: The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)

**Hypotheses**
Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genprndl.ord	⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧𝐺𝑥) <_Q (𝑧𝐺𝑦)))
genprndl.com	⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
genprndl.lower	⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑔𝐺ℎ) → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))

**Assertion**
Ref	Expression
genprndl	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))

Distinct variable groups: 𝑥,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣,𝑞,𝐴 𝑥,𝐵,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣,𝑞 𝑥,𝐺,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣,𝑞 𝑔,𝐹,𝑞 𝐴,𝑟,𝑞,𝑣,𝑤,𝑥,𝑦,𝑧 𝐵,𝑟,𝑔,ℎ ℎ,𝐹,𝑟,𝑣,𝑤,𝑥,𝑦,𝑧 𝐺,𝑟

Proof of Theorem genprndl Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		genpelvl.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2		genpelvl.2	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genpelvl 7873	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1^st ‘𝐴)∃𝑏 ∈ (1^st ‘𝐵)𝑞 = (𝑎𝐺𝑏)))
4		r2ex 2570	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (1^st ‘𝐴)∃𝑏 ∈ (1^st ‘𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
5	3, 4	bitrdi 196	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
6	5	biimpa 296	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))) → ∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
7	6	adantrl 482	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))) → ∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
8		prop 7836	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
9		prnmaxl 7849	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑎 ∈ (1^st ‘𝐴)) → ∃𝑐 ∈ (1^st ‘𝐴)𝑎 <_Q 𝑐)
10	8, 9	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝑎 ∈ (1^st ‘𝐴)) → ∃𝑐 ∈ (1^st ‘𝐴)𝑎 <_Q 𝑐)
11		prop 7836	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
12		prnmaxl 7849	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑏 ∈ (1^st ‘𝐵)) → ∃𝑑 ∈ (1^st ‘𝐵)𝑏 <_Q 𝑑)
13	11, 12	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ 𝑏 ∈ (1^st ‘𝐵)) → ∃𝑑 ∈ (1^st ‘𝐵)𝑏 <_Q 𝑑)
14	10, 13	anim12i 338	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝑎 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑏 ∈ (1^st ‘𝐵))) → (∃𝑐 ∈ (1^st ‘𝐴)𝑎 <_Q 𝑐 ∧ ∃𝑑 ∈ (1^st ‘𝐵)𝑏 <_Q 𝑑))
15	14	an4s 596	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵))) → (∃𝑐 ∈ (1^st ‘𝐴)𝑎 <_Q 𝑐 ∧ ∃𝑑 ∈ (1^st ‘𝐵)𝑏 <_Q 𝑑))
16		reeanv 2721	. . . . . . . . . . . . 13 ⊢ (∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎 <_Q 𝑐 ∧ 𝑏 <_Q 𝑑) ↔ (∃𝑐 ∈ (1^st ‘𝐴)𝑎 <_Q 𝑐 ∧ ∃𝑑 ∈ (1^st ‘𝐵)𝑏 <_Q 𝑑))
17	15, 16	sylibr 134	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵))) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎 <_Q 𝑐 ∧ 𝑏 <_Q 𝑑))
18		genprndl.ord	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧𝐺𝑥) <_Q (𝑧𝐺𝑦)))
19		genprndl.com	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
20	18, 19	genplt2i 7871	. . . . . . . . . . . . . 14 ⊢ ((𝑎 <_Q 𝑐 ∧ 𝑏 <_Q 𝑑) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
21	20	reximi 2647	. . . . . . . . . . . . 13 ⊢ (∃𝑑 ∈ (1^st ‘𝐵)(𝑎 <_Q 𝑐 ∧ 𝑏 <_Q 𝑑) → ∃𝑑 ∈ (1^st ‘𝐵)(𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
22	21	reximi 2647	. . . . . . . . . . . 12 ⊢ (∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎 <_Q 𝑐 ∧ 𝑏 <_Q 𝑑) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
23	17, 22	syl 14	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵))) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
24	23	adantrr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
25		breq1 4131	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝑎𝐺𝑏) → (𝑞 <_Q (𝑐𝐺𝑑) ↔ (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑)))
26	25	biimprd 158	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑎𝐺𝑏) → ((𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑) → 𝑞 <_Q (𝑐𝐺𝑑)))
27	26	reximdv 2651	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (1^st ‘𝐵)(𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑) → ∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑)))
28	27	reximdv 2651	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑)))
29	28	ad2antll 495	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → (∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)(𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑)))
30	24, 29	mpd 13	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑))
31	30	ex 115	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑)))
32	31	exlimdvv 1953	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑)))
33	32	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))) → (∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑)))
34	7, 33	mpd 13	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))) → ∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑))
35	1, 2	genpprecll 7875	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑐 ∈ (1^st ‘𝐴) ∧ 𝑑 ∈ (1^st ‘𝐵)) → (𝑐𝐺𝑑) ∈ (1^st ‘(𝐴𝐹𝐵))))
36	35	imp 124	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (1^st ‘𝐴) ∧ 𝑑 ∈ (1^st ‘𝐵))) → (𝑐𝐺𝑑) ∈ (1^st ‘(𝐴𝐹𝐵)))
37		elprnql 7842	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑐 ∈ (1^st ‘𝐴)) → 𝑐 ∈ Q)
38	8, 37	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑐 ∈ (1^st ‘𝐴)) → 𝑐 ∈ Q)
39		elprnql 7842	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑑 ∈ (1^st ‘𝐵)) → 𝑑 ∈ Q)
40	11, 39	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝑑 ∈ (1^st ‘𝐵)) → 𝑑 ∈ Q)
41	38, 40	anim12i 338	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑐 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑑 ∈ (1^st ‘𝐵))) → (𝑐 ∈ Q ∧ 𝑑 ∈ Q))
42	41	an4s 596	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (1^st ‘𝐴) ∧ 𝑑 ∈ (1^st ‘𝐵))) → (𝑐 ∈ Q ∧ 𝑑 ∈ Q))
43	2	caovcl 6238	. . . . . . . . . 10 ⊢ ((𝑐 ∈ Q ∧ 𝑑 ∈ Q) → (𝑐𝐺𝑑) ∈ Q)
44	42, 43	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (1^st ‘𝐴) ∧ 𝑑 ∈ (1^st ‘𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45		breq2 4132	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐𝐺𝑑) → (𝑞 <_Q 𝑟 ↔ 𝑞 <_Q (𝑐𝐺𝑑)))
46		eleq1 2301	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐𝐺𝑑) → (𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (1^st ‘(𝐴𝐹𝐵))))
47	45, 46	anbi12d 477	. . . . . . . . . 10 ⊢ (𝑟 = (𝑐𝐺𝑑) → ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <_Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1^st ‘(𝐴𝐹𝐵)))))
48	47	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (1^st ‘𝐴) ∧ 𝑑 ∈ (1^st ‘𝐵))) ∧ 𝑟 = (𝑐𝐺𝑑)) → ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <_Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1^st ‘(𝐴𝐹𝐵)))))
49	44, 48	rspcedv 2933	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (1^st ‘𝐴) ∧ 𝑑 ∈ (1^st ‘𝐵))) → ((𝑞 <_Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1^st ‘(𝐴𝐹𝐵))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))
50	36, 49	mpan2d 432	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (1^st ‘𝐴) ∧ 𝑑 ∈ (1^st ‘𝐵))) → (𝑞 <_Q (𝑐𝐺𝑑) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))
51	50	rexlimdvva 2676	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))
52	51	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))) → (∃𝑐 ∈ (1^st ‘𝐴)∃𝑑 ∈ (1^st ‘𝐵)𝑞 <_Q (𝑐𝐺𝑑) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))
53	34, 52	mpd 13	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵))))
54	53	expr 375	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ Q) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))
55		genprndl.lower	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑔𝐺ℎ) → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))
56	1, 2, 55	genpcdl 7880	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑥 <_Q 𝑟 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)))))
57	56	alrimdv 1929	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑥 <_Q 𝑟 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)))))
58		breq1 4131	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑞 → (𝑥 <_Q 𝑟 ↔ 𝑞 <_Q 𝑟))
59		eleq1 2301	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑞 → (𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
60	58, 59	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → ((𝑥 <_Q 𝑟 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <_Q 𝑟 → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))))
61	60	cbvalv 1973	. . . . . . . . 9 ⊢ (∀𝑥(𝑥 <_Q 𝑟 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))) ↔ ∀𝑞(𝑞 <_Q 𝑟 → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
62	57, 61	imbitrdi 161	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)) → ∀𝑞(𝑞 <_Q 𝑟 → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))))
63		sp 1564	. . . . . . . 8 ⊢ (∀𝑞(𝑞 <_Q 𝑟 → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))) → (𝑞 <_Q 𝑟 → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
64	62, 63	syl6 33	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑞 <_Q 𝑟 → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))))
65	64	impd 254	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 <_Q 𝑟) → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
66	65	ancomsd 269	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
67	66	ad2antrr 492	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ Q) ∧ 𝑟 ∈ Q) → ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
68	67	rexlimdva 2668	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
69	54, 68	impbid 129	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ Q) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))
70	69	ralrimiva 2623	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴𝐹𝐵)))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 ∀wal 1400 = wceq 1402 ∃wex 1545 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 {crab 2532 ⟨cop 3711 class class class wbr 4128 ‘cfv 5375 (class class class)co 6079 ∈ cmpo 6081 1^st c1st 6366 2^nd c2nd 6367 Qcnq 7641 <_Q cltq 7646 Pcnp 7652

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-eprel 4432 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-oadd 6685 df-omul 6686 df-er 6801 df-ec 6803 df-qs 6807 df-ni 7665 df-mi 7667 df-lti 7668 df-enq 7708 df-nqqs 7709 df-ltnqqs 7714 df-inp 7827

This theorem is referenced by: addclpr 7898 mulclpr 7933

W3C validator