Theorem gsumbagdiaglem 19577

Description: Lemma for gsumbagdiag 19578. (Contributed by Mario Carneiro, 5-Jan-2015.)

Hypotheses

Ref	Expression
psrbag.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
psrbagconf1o.1	⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹}
gsumbagdiag.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
gsumbagdiag.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)

Assertion

Ref	Expression
gsumbagdiaglem	⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑌 ∈ 𝑆 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)}))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐹   𝑥,𝑉,𝑦   𝑓,𝐼,𝑥,𝑦   𝑥,𝑆   𝑥,𝐷,𝑦   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐷(𝑓)   𝑆(𝑦,𝑓)   𝑉(𝑓)

Proof of Theorem gsumbagdiaglem

Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 813	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})
2		breq1 4807	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋) ↔ 𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)))
3	2	elrab 3504	. . . . 5 ⊢ (𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)} ↔ (𝑌 ∈ 𝐷 ∧ 𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)))
4	1, 3	sylib 208	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑌 ∈ 𝐷 ∧ 𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)))
5	4	simpld 477	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∈ 𝐷)
6	4	simprd 482	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋))
7		gsumbagdiag.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
8	7	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐼 ∈ 𝑉)
9		gsumbagdiag.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
10	9	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐹 ∈ 𝐷)
11		simprl 811	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∈ 𝑆)
12		breq1 4807	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 ∘𝑟 ≤ 𝐹 ↔ 𝑋 ∘𝑟 ≤ 𝐹))
13		psrbagconf1o.1	. . . . . . . . . 10 ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹}
14	12, 13	elrab2 3507	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ 𝐹))
15	11, 14	sylib 208	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ 𝐹))
16	15	simpld 477	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∈ 𝐷)
17		psrbag.d	. . . . . . . 8 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
18	17	psrbagf 19567	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
19	8, 16, 18	syl2anc 696	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋:𝐼⟶ℕ0)
20	15	simprd 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∘𝑟 ≤ 𝐹)
21	17	psrbagcon 19573	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝑋:𝐼⟶ℕ0 ∧ 𝑋 ∘𝑟 ≤ 𝐹)) → ((𝐹 ∘𝑓 − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹))
22	8, 10, 19, 20, 21	syl13anc 1479	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → ((𝐹 ∘𝑓 − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹))
23	22	simprd 482	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹)
24	17	psrbagf 19567	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
25	8, 5, 24	syl2anc 696	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌:𝐼⟶ℕ0)
26	22	simpld 477	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋) ∈ 𝐷)
27	17	psrbagf 19567	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∘𝑓 − 𝑋) ∈ 𝐷) → (𝐹 ∘𝑓 − 𝑋):𝐼⟶ℕ0)
28	8, 26, 27	syl2anc 696	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋):𝐼⟶ℕ0)
29	17	psrbagf 19567	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0)
30	8, 10, 29	syl2anc 696	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐹:𝐼⟶ℕ0)
31		nn0re 11493	. . . . . . 7 ⊢ (𝑢 ∈ ℕ0 → 𝑢 ∈ ℝ)
32		nn0re 11493	. . . . . . 7 ⊢ (𝑣 ∈ ℕ0 → 𝑣 ∈ ℝ)
33		nn0re 11493	. . . . . . 7 ⊢ (𝑤 ∈ ℕ0 → 𝑤 ∈ ℝ)
34		letr 10323	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤) → 𝑢 ≤ 𝑤))
35	31, 32, 33, 34	syl3an 1164	. . . . . 6 ⊢ ((𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0 ∧ 𝑤 ∈ ℕ0) → ((𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤) → 𝑢 ≤ 𝑤))
36	35	adantl 473	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ (𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0 ∧ 𝑤 ∈ ℕ0)) → ((𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤) → 𝑢 ≤ 𝑤))
37	8, 25, 28, 30, 36	caoftrn 7097	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → ((𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋) ∧ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹) → 𝑌 ∘𝑟 ≤ 𝐹))
38	6, 23, 37	mp2and 717	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∘𝑟 ≤ 𝐹)
39		breq1 4807	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∘𝑟 ≤ 𝐹 ↔ 𝑌 ∘𝑟 ≤ 𝐹))
40	39, 13	elrab2 3507	. . 3 ⊢ (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐷 ∧ 𝑌 ∘𝑟 ≤ 𝐹))
41	5, 38, 40	sylanbrc 701	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∈ 𝑆)
42	19	ffvelrnda 6522	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
43	25	ffvelrnda 6522	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
44	30	ffvelrnda 6522	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ ℕ0)
45		nn0re 11493	. . . . . . . 8 ⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℝ)
46		nn0re 11493	. . . . . . . 8 ⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℝ)
47		nn0re 11493	. . . . . . . 8 ⊢ ((𝐹‘𝑧) ∈ ℕ0 → (𝐹‘𝑧) ∈ ℝ)
48		leaddsub2 10697	. . . . . . . . 9 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑋‘𝑧) + (𝑌‘𝑧)) ≤ (𝐹‘𝑧) ↔ (𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧))))
49		leaddsub 10696	. . . . . . . . 9 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑋‘𝑧) + (𝑌‘𝑧)) ≤ (𝐹‘𝑧) ↔ (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
50	48, 49	bitr3d 270	. . . . . . . 8 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → ((𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧)) ↔ (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
51	45, 46, 47, 50	syl3an 1164	. . . . . . 7 ⊢ (((𝑋‘𝑧) ∈ ℕ0 ∧ (𝑌‘𝑧) ∈ ℕ0 ∧ (𝐹‘𝑧) ∈ ℕ0) → ((𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧)) ↔ (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
52	42, 43, 44, 51	syl3anc 1477	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧)) ↔ (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
53	52	ralbidva 3123	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (∀𝑧 ∈ 𝐼 (𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
54		ovexd 6843	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑋‘𝑧)) ∈ V)
55	25	feqmptd 6411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
56		ffn 6206	. . . . . . . 8 ⊢ (𝐹:𝐼⟶ℕ0 → 𝐹 Fn 𝐼)
57	30, 56	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐹 Fn 𝐼)
58		ffn 6206	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℕ0 → 𝑋 Fn 𝐼)
59	19, 58	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 Fn 𝐼)
60		inidm 3965	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
61		eqidd 2761	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) = (𝐹‘𝑧))
62		eqidd 2761	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) = (𝑋‘𝑧))
63	57, 59, 8, 8, 60, 61, 62	offval 7069	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑋‘𝑧))))
64	8, 43, 54, 55, 63	ofrfval2 7080	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋) ↔ ∀𝑧 ∈ 𝐼 (𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧))))
65		ovexd 6843	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑌‘𝑧)) ∈ V)
66	19	feqmptd 6411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
67		ffn 6206	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℕ0 → 𝑌 Fn 𝐼)
68	25, 67	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 Fn 𝐼)
69		eqidd 2761	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) = (𝑌‘𝑧))
70	57, 68, 8, 8, 60, 61, 69	offval 7069	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑌‘𝑧))))
71	8, 42, 65, 66, 70	ofrfval2 7080	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
72	53, 64, 71	3bitr4d 300	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋) ↔ 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)))
73	6, 72	mpbid 222	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌))
74		breq1 4807	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌) ↔ 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)))
75	74	elrab 3504	. . 3 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)} ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)))
76	16, 73, 75	sylanbrc 701	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)})
77	41, 76	jca 555	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑌 ∈ 𝑆 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  Vcvv 3340   class class class wbr 4804  ^◡ccnv 5265   “ cima 5269   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ∘_𝑓 cof 7060   ∘_𝑟 cofr 7061   ↑_𝑚 cmap 8023  Fincfn 8121  ℝcr 10127   + caddc 10131   ≤ cle 10267   − cmin 10458  ℕcn 11212  ℕ₀cn0 11484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485

This theorem is referenced by:  gsumbagdiag  19578  psrass1lem  19579