Theorem gsumbagdiaglem 19294

Description: Lemma for gsumbagdiag 19295. (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 795	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})
2		breq1 4616	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋) ↔ 𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)))
3	2	elrab 3346	. . . . 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 475	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∈ 𝐷)
6	4	simprd 479	. . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐼 ∈ 𝑉)
9		gsumbagdiag.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐹 ∈ 𝐷)
11		simprl 793	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∈ 𝑆)
12		breq1 4616	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 ∘𝑟 ≤ 𝐹 ↔ 𝑋 ∘𝑟 ≤ 𝐹))
13		psrbagconf1o.1	. . . . . . . . . 10 ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹}
14	12, 13	elrab2 3348	. . . . . . . . 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 475	. . . . . . 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 19284	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
19	8, 16, 18	syl2anc 692	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋:𝐼⟶ℕ0)
20	15	simprd 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∘𝑟 ≤ 𝐹)
21	17	psrbagcon 19290	. . . . . 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 1325	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → ((𝐹 ∘𝑓 − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹))
23	22	simprd 479	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹)
24	17	psrbagf 19284	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
25	8, 5, 24	syl2anc 692	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌:𝐼⟶ℕ0)
26	22	simpld 475	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋) ∈ 𝐷)
27	17	psrbagf 19284	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∘𝑓 − 𝑋) ∈ 𝐷) → (𝐹 ∘𝑓 − 𝑋):𝐼⟶ℕ0)
28	8, 26, 27	syl2anc 692	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋):𝐼⟶ℕ0)
29	17	psrbagf 19284	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0)
30	8, 10, 29	syl2anc 692	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐹:𝐼⟶ℕ0)
31		nn0re 11245	. . . . . . 7 ⊢ (𝑢 ∈ ℕ0 → 𝑢 ∈ ℝ)
32		nn0re 11245	. . . . . . 7 ⊢ (𝑣 ∈ ℕ0 → 𝑣 ∈ ℝ)
33		nn0re 11245	. . . . . . 7 ⊢ (𝑤 ∈ ℕ0 → 𝑤 ∈ ℝ)
34		letr 10075	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤) → 𝑢 ≤ 𝑤))
35	31, 32, 33, 34	syl3an 1365	. . . . . 6 ⊢ ((𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0 ∧ 𝑤 ∈ ℕ0) → ((𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤) → 𝑢 ≤ 𝑤))
36	35	adantl 482	. . . . 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 6885	. . . 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 714	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∘𝑟 ≤ 𝐹)
39		breq1 4616	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∘𝑟 ≤ 𝐹 ↔ 𝑌 ∘𝑟 ≤ 𝐹))
40	39, 13	elrab2 3348	. . 3 ⊢ (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐷 ∧ 𝑌 ∘𝑟 ≤ 𝐹))
41	5, 38, 40	sylanbrc 697	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 ∈ 𝑆)
42	19	ffvelrnda 6315	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
43	25	ffvelrnda 6315	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
44	30	ffvelrnda 6315	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ ℕ0)
45		nn0re 11245	. . . . . . . 8 ⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℝ)
46		nn0re 11245	. . . . . . . 8 ⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℝ)
47		nn0re 11245	. . . . . . . 8 ⊢ ((𝐹‘𝑧) ∈ ℕ0 → (𝐹‘𝑧) ∈ ℝ)
48		leaddsub2 10449	. . . . . . . . 9 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑋‘𝑧) + (𝑌‘𝑧)) ≤ (𝐹‘𝑧) ↔ (𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧))))
49		leaddsub 10448	. . . . . . . . 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 1365	. . . . . . 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 1323	. . . . . 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 2979	. . . . 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		ovex 6632	. . . . . . 7 ⊢ ((𝐹‘𝑧) − (𝑋‘𝑧)) ∈ V
55	54	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑋‘𝑧)) ∈ V)
56	25	feqmptd 6206	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
57		ffn 6002	. . . . . . . 8 ⊢ (𝐹:𝐼⟶ℕ0 → 𝐹 Fn 𝐼)
58	30, 57	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝐹 Fn 𝐼)
59		ffn 6002	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℕ0 → 𝑋 Fn 𝐼)
60	19, 59	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 Fn 𝐼)
61		inidm 3800	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
62		eqidd 2622	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) = (𝐹‘𝑧))
63		eqidd 2622	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) = (𝑋‘𝑧))
64	58, 60, 8, 8, 61, 62, 63	offval 6857	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑋‘𝑧))))
65	8, 43, 55, 56, 64	ofrfval2 6868	. . . . 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))))
66		ovex 6632	. . . . . . 7 ⊢ ((𝐹‘𝑧) − (𝑌‘𝑧)) ∈ V
67	66	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑌‘𝑧)) ∈ V)
68	19	feqmptd 6206	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
69		ffn 6002	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℕ0 → 𝑌 Fn 𝐼)
70	25, 69	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑌 Fn 𝐼)
71		eqidd 2622	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) = (𝑌‘𝑧))
72	58, 70, 8, 8, 61, 62, 71	offval 6857	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝐹 ∘𝑓 − 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑌‘𝑧))))
73	8, 42, 67, 68, 72	ofrfval2 6868	. . . . 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))))
74	53, 65, 73	3bitr4d 300	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → (𝑌 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋) ↔ 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)))
75	6, 74	mpbid 222	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌))
76		breq1 4616	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌) ↔ 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)))
77	76	elrab 3346	. . 3 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)} ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)))
78	16, 75, 77	sylanbrc 697	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑋)})) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝑌)})
79	41, 78	jca 554	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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911  Vcvv 3186   class class class wbr 4613  ^◡ccnv 5073   “ cima 5077   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ∘_𝑓 cof 6848   ∘_𝑟 cofr 6849   ↑_𝑚 cmap 7802  Fincfn 7899  ℝcr 9879   + caddc 9883   ≤ cle 10019   − cmin 10210  ℕcn 10964  ℕ₀cn0 11236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237

This theorem is referenced by:  gsumbagdiag  19295  psrass1lem  19296