MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumbagdiaglem Structured version   Visualization version   GIF version

Theorem gsumbagdiaglem 20083
Description: Lemma for gsumbagdiag 20084. (Contributed by Mario Carneiro, 5-Jan-2015.)
Hypotheses
Ref Expression
psrbag.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrbagconf1o.1 𝑆 = {𝑦𝐷𝑦r𝐹}
gsumbagdiag.i (𝜑𝐼𝑉)
gsumbagdiag.f (𝜑𝐹𝐷)
Assertion
Ref Expression
gsumbagdiaglem ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑌)}))
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.
StepHypRef Expression
1 simprr 769 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})
2 breq1 5060 . . . . . 6 (𝑥 = 𝑌 → (𝑥r ≤ (𝐹f𝑋) ↔ 𝑌r ≤ (𝐹f𝑋)))
32elrab 3677 . . . . 5 (𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)} ↔ (𝑌𝐷𝑌r ≤ (𝐹f𝑋)))
41, 3sylib 219 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌𝐷𝑌r ≤ (𝐹f𝑋)))
54simpld 495 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌𝐷)
64simprd 496 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌r ≤ (𝐹f𝑋))
7 gsumbagdiag.i . . . . . . 7 (𝜑𝐼𝑉)
87adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐼𝑉)
9 gsumbagdiag.f . . . . . . 7 (𝜑𝐹𝐷)
109adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐹𝐷)
11 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋𝑆)
12 breq1 5060 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑦r𝐹𝑋r𝐹))
13 psrbagconf1o.1 . . . . . . . . . 10 𝑆 = {𝑦𝐷𝑦r𝐹}
1412, 13elrab2 3680 . . . . . . . . 9 (𝑋𝑆 ↔ (𝑋𝐷𝑋r𝐹))
1511, 14sylib 219 . . . . . . . 8 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑋𝐷𝑋r𝐹))
1615simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋𝐷)
17 psrbag.d . . . . . . . 8 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
1817psrbagf 20073 . . . . . . 7 ((𝐼𝑉𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
198, 16, 18syl2anc 584 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋:𝐼⟶ℕ0)
2015simprd 496 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋r𝐹)
2117psrbagcon 20079 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝐷𝑋:𝐼⟶ℕ0𝑋r𝐹)) → ((𝐹f𝑋) ∈ 𝐷 ∧ (𝐹f𝑋) ∘r𝐹))
228, 10, 19, 20, 21syl13anc 1364 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → ((𝐹f𝑋) ∈ 𝐷 ∧ (𝐹f𝑋) ∘r𝐹))
2322simprd 496 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋) ∘r𝐹)
2417psrbagf 20073 . . . . . 6 ((𝐼𝑉𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
258, 5, 24syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌:𝐼⟶ℕ0)
2622simpld 495 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋) ∈ 𝐷)
2717psrbagf 20073 . . . . . 6 ((𝐼𝑉 ∧ (𝐹f𝑋) ∈ 𝐷) → (𝐹f𝑋):𝐼⟶ℕ0)
288, 26, 27syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋):𝐼⟶ℕ0)
2917psrbagf 20073 . . . . . 6 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
308, 10, 29syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐹:𝐼⟶ℕ0)
31 nn0re 11894 . . . . . . 7 (𝑢 ∈ ℕ0𝑢 ∈ ℝ)
32 nn0re 11894 . . . . . . 7 (𝑣 ∈ ℕ0𝑣 ∈ ℝ)
33 nn0re 11894 . . . . . . 7 (𝑤 ∈ ℕ0𝑤 ∈ ℝ)
34 letr 10722 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3531, 32, 33, 34syl3an 1152 . . . . . 6 ((𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3635adantl 482 . . . . 5 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ (𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0)) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
378, 25, 28, 30, 36caoftrn 7433 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → ((𝑌r ≤ (𝐹f𝑋) ∧ (𝐹f𝑋) ∘r𝐹) → 𝑌r𝐹))
386, 23, 37mp2and 695 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌r𝐹)
39 breq1 5060 . . . 4 (𝑦 = 𝑌 → (𝑦r𝐹𝑌r𝐹))
4039, 13elrab2 3680 . . 3 (𝑌𝑆 ↔ (𝑌𝐷𝑌r𝐹))
415, 38, 40sylanbrc 583 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌𝑆)
42 breq1 5060 . . 3 (𝑥 = 𝑋 → (𝑥r ≤ (𝐹f𝑌) ↔ 𝑋r ≤ (𝐹f𝑌)))
4319ffvelrnda 6843 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
4425ffvelrnda 6843 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
4530ffvelrnda 6843 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
46 nn0re 11894 . . . . . . . 8 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℝ)
47 nn0re 11894 . . . . . . . 8 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℝ)
48 nn0re 11894 . . . . . . . 8 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℝ)
49 leaddsub2 11105 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
50 leaddsub 11104 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5149, 50bitr3d 282 . . . . . . . 8 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5246, 47, 48, 51syl3an 1152 . . . . . . 7 (((𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝐹𝑧) ∈ ℕ0) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5343, 44, 45, 52syl3anc 1363 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5453ralbidva 3193 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
55 ovexd 7180 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑋𝑧)) ∈ V)
5625feqmptd 6726 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
5730ffnd 6508 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐹 Fn 𝐼)
5819ffnd 6508 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋 Fn 𝐼)
59 inidm 4192 . . . . . . 7 (𝐼𝐼) = 𝐼
60 eqidd 2819 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) = (𝐹𝑧))
61 eqidd 2819 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) = (𝑋𝑧))
6257, 58, 8, 8, 59, 60, 61offval 7405 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑋𝑧))))
638, 44, 55, 56, 62ofrfval2 7416 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌r ≤ (𝐹f𝑋) ↔ ∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
64 ovexd 7180 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑌𝑧)) ∈ V)
6519feqmptd 6726 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
6625ffnd 6508 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌 Fn 𝐼)
67 eqidd 2819 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) = (𝑌𝑧))
6857, 66, 8, 8, 59, 60, 67offval 7405 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑌) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑌𝑧))))
698, 43, 64, 65, 68ofrfval2 7416 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑋r ≤ (𝐹f𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
7054, 63, 693bitr4d 312 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌r ≤ (𝐹f𝑋) ↔ 𝑋r ≤ (𝐹f𝑌)))
716, 70mpbid 233 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋r ≤ (𝐹f𝑌))
7242, 16, 71elrabd 3679 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑌)})
7341, 72jca 512 1 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492   class class class wbr 5057  ccnv 5547  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396  r cofr 7397  m cmap 8395  Fincfn 8497  cr 10524   + caddc 10528  cle 10664  cmin 10858  cn 11626  0cn0 11885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886
This theorem is referenced by:  gsumbagdiag  20084  psrass1lem  20085
  Copyright terms: Public domain W3C validator