Theorem gsumbagdiaglemOLD 20754

Description: Obsolete version of gsumbagdiaglem 20757 as of 6-Aug-2024. (Contributed by Mario Carneiro, 5-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
gsumbagdiaglemOLD	⊢ ((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 gsumbagdiaglemOLD

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

Step	Hyp	Ref	Expression
1		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})
2		breq1 5034	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 ∘r ≤ (𝐹 ∘f − 𝑋) ↔ 𝑌 ∘r ≤ (𝐹 ∘f − 𝑋)))
3	2	elrab 3589	. . . . 5 ⊢ (𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)} ↔ (𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ (𝐹 ∘f − 𝑋)))
4	1, 3	sylib 221	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ (𝐹 ∘f − 𝑋)))
5	4	simpld 498	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 ∈ 𝐷)
6	4	simprd 499	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 ∘r ≤ (𝐹 ∘f − 𝑋))
7		gsumbagdiagOLD.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
8	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐼 ∈ 𝑉)
9		gsumbagdiagOLD.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
10	9	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐹 ∈ 𝐷)
11		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∈ 𝑆)
12		breq1 5034	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 ∘r ≤ 𝐹 ↔ 𝑋 ∘r ≤ 𝐹))
13		psrbagconf1o.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}
14	12, 13	elrab2 3592	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹))
15	11, 14	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹))
16	15	simpld 498	. . . . . . 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	psrbagfOLD 20735	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
19	8, 16, 18	syl2anc 587	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋:𝐼⟶ℕ0)
20	15	simprd 499	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∘r ≤ 𝐹)
21	17	psrbagconOLD 20747	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → ((𝐹 ∘f − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘f − 𝑋) ∘r ≤ 𝐹))
23	22	simprd 499	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋) ∘r ≤ 𝐹)
24	17	psrbagfOLD 20735	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
25	8, 5, 24	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌:𝐼⟶ℕ0)
26	22	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋) ∈ 𝐷)
27	17	psrbagfOLD 20735	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∘f − 𝑋) ∈ 𝐷) → (𝐹 ∘f − 𝑋):𝐼⟶ℕ0)
28	8, 26, 27	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋):𝐼⟶ℕ0)
29	17	psrbagfOLD 20735	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0)
30	8, 10, 29	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐹:𝐼⟶ℕ0)
31		nn0re 11988	. . . . . . 7 ⊢ (𝑢 ∈ ℕ0 → 𝑢 ∈ ℝ)
32		nn0re 11988	. . . . . . 7 ⊢ (𝑣 ∈ ℕ0 → 𝑣 ∈ ℝ)
33		nn0re 11988	. . . . . . 7 ⊢ (𝑤 ∈ ℕ0 → 𝑤 ∈ ℝ)
34		letr 10815	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤) → 𝑢 ≤ 𝑤))
35	31, 32, 33, 34	syl3an 1161	. . . . . 6 ⊢ ((𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0 ∧ 𝑤 ∈ ℕ0) → ((𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤) → 𝑢 ≤ 𝑤))
36	35	adantl 485	. . . . 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 7465	. . . 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 699	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 ∘r ≤ 𝐹)
39		breq1 5034	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∘r ≤ 𝐹 ↔ 𝑌 ∘r ≤ 𝐹))
40	39, 13	elrab2 3592	. . 3 ⊢ (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ 𝐹))
41	5, 38, 40	sylanbrc 586	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 ∈ 𝑆)
42		breq1 5034	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ∘r ≤ (𝐹 ∘f − 𝑌) ↔ 𝑋 ∘r ≤ (𝐹 ∘f − 𝑌)))
43	19	ffvelrnda 6864	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
44	25	ffvelrnda 6864	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
45	30	ffvelrnda 6864	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ ℕ0)
46		nn0re 11988	. . . . . . . 8 ⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℝ)
47		nn0re 11988	. . . . . . . 8 ⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℝ)
48		nn0re 11988	. . . . . . . 8 ⊢ ((𝐹‘𝑧) ∈ ℕ0 → (𝐹‘𝑧) ∈ ℝ)
49		leaddsub2 11198	. . . . . . . . 9 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑋‘𝑧) + (𝑌‘𝑧)) ≤ (𝐹‘𝑧) ↔ (𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧))))
50		leaddsub 11197	. . . . . . . . 9 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑋‘𝑧) + (𝑌‘𝑧)) ≤ (𝐹‘𝑧) ↔ (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
51	49, 50	bitr3d 284	. . . . . . . 8 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → ((𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧)) ↔ (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
52	46, 47, 48, 51	syl3an 1161	. . . . . . 7 ⊢ (((𝑋‘𝑧) ∈ ℕ0 ∧ (𝑌‘𝑧) ∈ ℕ0 ∧ (𝐹‘𝑧) ∈ ℕ0) → ((𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧)) ↔ (𝑋‘𝑧) ≤ ((𝐹‘𝑧) − (𝑌‘𝑧))))
53	43, 44, 45, 52	syl3anc 1372	. . . . . 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))))
54	53	ralbidva 3109	. . . . 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))))
55		ovexd 7208	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑋‘𝑧)) ∈ V)
56	25	feqmptd 6740	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
57	30	ffnd 6506	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐹 Fn 𝐼)
58	19	ffnd 6506	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 Fn 𝐼)
59		inidm 4110	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
60		eqidd 2740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) = (𝐹‘𝑧))
61		eqidd 2740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) = (𝑋‘𝑧))
62	57, 58, 8, 8, 59, 60, 61	offval 7436	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑋‘𝑧))))
63	8, 44, 55, 56, 62	ofrfval2 7448	. . . . 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))))
64		ovexd 7208	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑌‘𝑧)) ∈ V)
65	19	feqmptd 6740	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
66	25	ffnd 6506	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 Fn 𝐼)
67		eqidd 2740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) = (𝑌‘𝑧))
68	57, 66, 8, 8, 59, 60, 67	offval 7436	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑌‘𝑧))))
69	8, 43, 64, 65, 68	ofrfval2 7448	. . . . 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))))
70	54, 63, 69	3bitr4d 314	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑌 ∘r ≤ (𝐹 ∘f − 𝑋) ↔ 𝑋 ∘r ≤ (𝐹 ∘f − 𝑌)))
71	6, 70	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∘r ≤ (𝐹 ∘f − 𝑌))
72	42, 16, 71	elrabd 3591	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑌)})
73	41, 72	jca 515	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 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3054  {crab 3058  Vcvv 3399   class class class wbr 5031  ^◡ccnv 5525   “ cima 5529  ⟶wf 6336  ‘cfv 6340  (class class class)co 7173   ∘_f cof 7426   ∘_r cofr 7427   ↑_m cmap 8440  Fincfn 8558  ℝcr 10617   + caddc 10621   ≤ cle 10757   − cmin 10951  ℕcn 11719  ℕ₀cn0 11979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-cnex 10674  ax-resscn 10675  ax-1cn 10676  ax-icn 10677  ax-addcl 10678  ax-addrcl 10679  ax-mulcl 10680  ax-mulrcl 10681  ax-mulcom 10682  ax-addass 10683  ax-mulass 10684  ax-distr 10685  ax-i2m1 10686  ax-1ne0 10687  ax-1rid 10688  ax-rnegex 10689  ax-rrecex 10690  ax-cnre 10691  ax-pre-lttri 10692  ax-pre-lttrn 10693  ax-pre-ltadd 10694  ax-pre-mulgt0 10695

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7130  df-ov 7176  df-oprab 7177  df-mpo 7178  df-of 7428  df-ofr 7429  df-om 7603  df-supp 7860  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-1o 8134  df-er 8323  df-map 8442  df-en 8559  df-dom 8560  df-sdom 8561  df-fin 8562  df-pnf 10758  df-mnf 10759  df-xr 10760  df-ltxr 10761  df-le 10762  df-sub 10953  df-neg 10954  df-nn 11720  df-n0 11980

This theorem is referenced by:  gsumbagdiagOLD  20755  psrass1lemOLD  20756