Theorem gsumbagdiaglem 20754

Description: Lemma for gsumbagdiag 20755. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 6-Aug-2024.)

Hypotheses

Ref	Expression
gsumbagdiag.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
gsumbagdiag.s	⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}
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,𝐷   𝑦,𝐷   𝑓,𝐹   𝑥,𝐹   𝑦,𝐹   𝑓,𝐼   𝑓,𝑋   𝑥,𝑋   𝑦,𝑋   𝑓,𝑌   𝑥,𝑌   𝑦,𝑌

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

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 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})
2		breq1 5033	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 ∘r ≤ (𝐹 ∘f − 𝑋) ↔ 𝑌 ∘r ≤ (𝐹 ∘f − 𝑋)))
3	2	elrab 3588	. . . . 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		gsumbagdiag.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
8	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐹 ∈ 𝐷)
9		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∈ 𝑆)
10		breq1 5033	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (𝑦 ∘r ≤ 𝐹 ↔ 𝑋 ∘r ≤ 𝐹))
11		gsumbagdiag.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}
12	10, 11	elrab2 3591	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹))
13	9, 12	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹))
14	13	simpld 498	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∈ 𝐷)
15		gsumbagdiag.d	. . . . . . . 8 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
16	15	psrbagf 20731	. . . . . . 7 ⊢ (𝑋 ∈ 𝐷 → 𝑋:𝐼⟶ℕ0)
17	14, 16	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋:𝐼⟶ℕ0)
18	13	simprd 499	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∘r ≤ 𝐹)
19	15	psrbagcon 20743	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑋:𝐼⟶ℕ0 ∧ 𝑋 ∘r ≤ 𝐹) → ((𝐹 ∘f − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘f − 𝑋) ∘r ≤ 𝐹))
20	8, 17, 18, 19	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → ((𝐹 ∘f − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘f − 𝑋) ∘r ≤ 𝐹))
21	20	simprd 499	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋) ∘r ≤ 𝐹)
22	15	psrbagf 20731	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
23	8, 22	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐹:𝐼⟶ℕ0)
24	23	ffnd 6505	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐹 Fn 𝐼)
25	8, 24	fndmexd 7637	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝐼 ∈ V)
26	15	psrbagf 20731	. . . . . 6 ⊢ (𝑌 ∈ 𝐷 → 𝑌:𝐼⟶ℕ0)
27	5, 26	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌:𝐼⟶ℕ0)
28	20	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋) ∈ 𝐷)
29	15	psrbagf 20731	. . . . . 6 ⊢ ((𝐹 ∘f − 𝑋) ∈ 𝐷 → (𝐹 ∘f − 𝑋):𝐼⟶ℕ0)
30	28, 29	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋):𝐼⟶ℕ0)
31		nn0re 11985	. . . . . . 7 ⊢ (𝑢 ∈ ℕ0 → 𝑢 ∈ ℝ)
32		nn0re 11985	. . . . . . 7 ⊢ (𝑣 ∈ ℕ0 → 𝑣 ∈ ℝ)
33		nn0re 11985	. . . . . . 7 ⊢ (𝑤 ∈ ℕ0 → 𝑤 ∈ ℝ)
34		letr 10812	. . . . . . 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	25, 27, 30, 23, 36	caoftrn 7462	. . . 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, 21, 37	mp2and 699	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 ∘r ≤ 𝐹)
39		breq1 5033	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∘r ≤ 𝐹 ↔ 𝑌 ∘r ≤ 𝐹))
40	39, 11	elrab2 3591	. . 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 5033	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ∘r ≤ (𝐹 ∘f − 𝑌) ↔ 𝑋 ∘r ≤ (𝐹 ∘f − 𝑌)))
43	17	ffvelrnda 6861	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
44	27	ffvelrnda 6861	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
45	23	ffvelrnda 6861	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ ℕ0)
46		nn0re 11985	. . . . . . . 8 ⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℝ)
47		nn0re 11985	. . . . . . . 8 ⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℝ)
48		nn0re 11985	. . . . . . . 8 ⊢ ((𝐹‘𝑧) ∈ ℕ0 → (𝐹‘𝑧) ∈ ℝ)
49		leaddsub2 11195	. . . . . . . . 9 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑋‘𝑧) + (𝑌‘𝑧)) ≤ (𝐹‘𝑧) ↔ (𝑌‘𝑧) ≤ ((𝐹‘𝑧) − (𝑋‘𝑧))))
50		leaddsub 11194	. . . . . . . . 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 3108	. . . . 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 7205	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑋‘𝑧)) ∈ V)
56	27	feqmptd 6737	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
57	17	ffnd 6505	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 Fn 𝐼)
58		inidm 4109	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
59		eqidd 2739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) = (𝐹‘𝑧))
60		eqidd 2739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) = (𝑋‘𝑧))
61	24, 57, 25, 25, 58, 59, 60	offval 7433	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑋‘𝑧))))
62	25, 44, 55, 56, 61	ofrfval2 7445	. . . . 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))))
63		ovexd 7205	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧) − (𝑌‘𝑧)) ∈ V)
64	17	feqmptd 6737	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
65	27	ffnd 6505	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑌 Fn 𝐼)
66		eqidd 2739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) = (𝑌‘𝑧))
67	24, 65, 25, 25, 58, 59, 66	offval 7433	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝐹 ∘f − 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝐹‘𝑧) − (𝑌‘𝑧))))
68	25, 43, 63, 64, 67	ofrfval2 7445	. . . . 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))))
69	54, 62, 68	3bitr4d 314	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑌 ∘r ≤ (𝐹 ∘f − 𝑋) ↔ 𝑋 ∘r ≤ (𝐹 ∘f − 𝑌)))
70	6, 69	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∘r ≤ (𝐹 ∘f − 𝑌))
71	42, 14, 70	elrabd 3590	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑌)})
72	41, 71	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 3053  {crab 3057  Vcvv 3398   class class class wbr 5030  ^◡ccnv 5524   “ cima 5528  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170   ∘_f cof 7423   ∘_r cofr 7424   ↑_m cmap 8437  Fincfn 8555  ℝcr 10614   + caddc 10618   ≤ cle 10754   − cmin 10948  ℕcn 11716  ℕ₀cn0 11976

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 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692

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 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-ofr 7426  df-om 7600  df-1st 7714  df-2nd 7715  df-supp 7857  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-n0 11977

This theorem is referenced by:  gsumbagdiag  20755  psrass1lem  20756