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Theorem gsumbagdiaglem 21343
Description: Lemma for gsumbagdiag 21344. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 6-Aug-2024.)
Hypotheses
Ref Expression
gsumbagdiag.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
gsumbagdiag.s 𝑆 = {𝑦𝐷𝑦r𝐹}
gsumbagdiag.f (𝜑𝐹𝐷)
Assertion
Ref Expression
gsumbagdiaglem ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑌)}))
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.
StepHypRef Expression
1 simprr 771 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})
2 breq1 5108 . . . . . 6 (𝑥 = 𝑌 → (𝑥r ≤ (𝐹f𝑋) ↔ 𝑌r ≤ (𝐹f𝑋)))
32elrab 3645 . . . . 5 (𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)} ↔ (𝑌𝐷𝑌r ≤ (𝐹f𝑋)))
41, 3sylib 217 . . . 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.f . . . . . . 7 (𝜑𝐹𝐷)
87adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐹𝐷)
9 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋𝑆)
10 breq1 5108 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑦r𝐹𝑋r𝐹))
11 gsumbagdiag.s . . . . . . . . . 10 𝑆 = {𝑦𝐷𝑦r𝐹}
1210, 11elrab2 3648 . . . . . . . . 9 (𝑋𝑆 ↔ (𝑋𝐷𝑋r𝐹))
139, 12sylib 217 . . . . . . . 8 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑋𝐷𝑋r𝐹))
1413simpld 495 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋𝐷)
15 gsumbagdiag.d . . . . . . . 8 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
1615psrbagf 21320 . . . . . . 7 (𝑋𝐷𝑋:𝐼⟶ℕ0)
1714, 16syl 17 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋:𝐼⟶ℕ0)
1813simprd 496 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋r𝐹)
1915psrbagcon 21332 . . . . . 6 ((𝐹𝐷𝑋:𝐼⟶ℕ0𝑋r𝐹) → ((𝐹f𝑋) ∈ 𝐷 ∧ (𝐹f𝑋) ∘r𝐹))
208, 17, 18, 19syl3anc 1371 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → ((𝐹f𝑋) ∈ 𝐷 ∧ (𝐹f𝑋) ∘r𝐹))
2120simprd 496 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋) ∘r𝐹)
2215psrbagf 21320 . . . . . . . 8 (𝐹𝐷𝐹:𝐼⟶ℕ0)
238, 22syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐹:𝐼⟶ℕ0)
2423ffnd 6669 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐹 Fn 𝐼)
258, 24fndmexd 7843 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝐼 ∈ V)
2615psrbagf 21320 . . . . . 6 (𝑌𝐷𝑌:𝐼⟶ℕ0)
275, 26syl 17 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌:𝐼⟶ℕ0)
2820simpld 495 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋) ∈ 𝐷)
2915psrbagf 21320 . . . . . 6 ((𝐹f𝑋) ∈ 𝐷 → (𝐹f𝑋):𝐼⟶ℕ0)
3028, 29syl 17 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋):𝐼⟶ℕ0)
31 nn0re 12422 . . . . . . 7 (𝑢 ∈ ℕ0𝑢 ∈ ℝ)
32 nn0re 12422 . . . . . . 7 (𝑣 ∈ ℕ0𝑣 ∈ ℝ)
33 nn0re 12422 . . . . . . 7 (𝑤 ∈ ℕ0𝑤 ∈ ℝ)
34 letr 11249 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3531, 32, 33, 34syl3an 1160 . . . . . 6 ((𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3635adantl 482 . . . . 5 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ (𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0)) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3725, 27, 30, 23, 36caoftrn 7655 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → ((𝑌r ≤ (𝐹f𝑋) ∧ (𝐹f𝑋) ∘r𝐹) → 𝑌r𝐹))
386, 21, 37mp2and 697 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌r𝐹)
39 breq1 5108 . . . 4 (𝑦 = 𝑌 → (𝑦r𝐹𝑌r𝐹))
4039, 11elrab2 3648 . . 3 (𝑌𝑆 ↔ (𝑌𝐷𝑌r𝐹))
415, 38, 40sylanbrc 583 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌𝑆)
42 breq1 5108 . . 3 (𝑥 = 𝑋 → (𝑥r ≤ (𝐹f𝑌) ↔ 𝑋r ≤ (𝐹f𝑌)))
4317ffvelcdmda 7035 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
4427ffvelcdmda 7035 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
4523ffvelcdmda 7035 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
46 nn0re 12422 . . . . . . . 8 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℝ)
47 nn0re 12422 . . . . . . . 8 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℝ)
48 nn0re 12422 . . . . . . . 8 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℝ)
49 leaddsub2 11632 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
50 leaddsub 11631 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5149, 50bitr3d 280 . . . . . . . 8 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5246, 47, 48, 51syl3an 1160 . . . . . . 7 (((𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝐹𝑧) ∈ ℕ0) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5343, 44, 45, 52syl3anc 1371 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5453ralbidva 3172 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
55 ovexd 7392 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑋𝑧)) ∈ V)
5627feqmptd 6910 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
5717ffnd 6669 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋 Fn 𝐼)
58 inidm 4178 . . . . . . 7 (𝐼𝐼) = 𝐼
59 eqidd 2737 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) = (𝐹𝑧))
60 eqidd 2737 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) = (𝑋𝑧))
6124, 57, 25, 25, 58, 59, 60offval 7626 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑋) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑋𝑧))))
6225, 44, 55, 56, 61ofrfval2 7638 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌r ≤ (𝐹f𝑋) ↔ ∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
63 ovexd 7392 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑌𝑧)) ∈ V)
6417feqmptd 6910 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
6527ffnd 6669 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑌 Fn 𝐼)
66 eqidd 2737 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) = (𝑌𝑧))
6724, 65, 25, 25, 58, 59, 66offval 7626 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝐹f𝑌) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑌𝑧))))
6825, 43, 63, 64, 67ofrfval2 7638 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑋r ≤ (𝐹f𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
6954, 62, 683bitr4d 310 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌r ≤ (𝐹f𝑋) ↔ 𝑋r ≤ (𝐹f𝑌)))
706, 69mpbid 231 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋r ≤ (𝐹f𝑌))
7142, 14, 70elrabd 3647 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → 𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑌)})
7241, 71jca 512 1 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445   class class class wbr 5105  ccnv 5632  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  r cofr 7616  m cmap 8765  Fincfn 8883  cr 11050   + caddc 11054  cle 11190  cmin 11385  cn 12153  0cn0 12413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414
This theorem is referenced by:  gsumbagdiag  21344  psrass1lem  21345
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