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Theorem cantnflem1d 8835
Description: Lemma for cantnf 8840. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
oemapval.f (𝜑𝐹𝑆)
oemapval.g (𝜑𝐺𝑆)
oemapvali.r (𝜑𝐹𝑇𝐺)
oemapvali.x 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
cantnflem1.o 𝑂 = OrdIso( E , (𝐺 supp ∅))
cantnflem1.h 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐺‘(𝑂𝑘))) +𝑜 𝑧)), ∅)
Assertion
Ref Expression
cantnflem1d (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ (𝐻‘suc (𝑂𝑋)))
Distinct variable groups:   𝑘,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑘,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑘   𝑘,𝐹,𝑤,𝑥,𝑦,𝑧   𝑆,𝑐,𝑘,𝑥,𝑦,𝑧   𝐺,𝑐,𝑘,𝑤,𝑥,𝑦,𝑧   𝑥,𝐻,𝑦   𝑘,𝑂,𝑤,𝑥,𝑦,𝑧   𝜑,𝑘,𝑥,𝑦,𝑧   𝑘,𝑋,𝑤,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝐻(𝑧,𝑤,𝑘,𝑐)   𝑂(𝑐)   𝑋(𝑐)

Proof of Theorem cantnflem1d
StepHypRef Expression
1 cantnfs.a . . . . . 6 (𝜑𝐴 ∈ On)
2 cantnfs.b . . . . . . 7 (𝜑𝐵 ∈ On)
3 cantnfs.s . . . . . . . . 9 𝑆 = dom (𝐴 CNF 𝐵)
4 oemapval.t . . . . . . . . 9 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
5 oemapval.f . . . . . . . . 9 (𝜑𝐹𝑆)
6 oemapval.g . . . . . . . . 9 (𝜑𝐺𝑆)
7 oemapvali.r . . . . . . . . 9 (𝜑𝐹𝑇𝐺)
8 oemapvali.x . . . . . . . . 9 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
93, 1, 2, 4, 5, 6, 7, 8oemapvali 8831 . . . . . . . 8 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
109simp1d 1165 . . . . . . 7 (𝜑𝑋𝐵)
11 onelon 5968 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
122, 10, 11syl2anc 575 . . . . . 6 (𝜑𝑋 ∈ On)
13 oecl 7857 . . . . . 6 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
141, 12, 13syl2anc 575 . . . . 5 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
153, 1, 2cantnfs 8813 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
166, 15mpbid 223 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 484 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
1817, 10ffvelrnd 6585 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
19 onelon 5968 . . . . . 6 ((𝐴 ∈ On ∧ (𝐺𝑋) ∈ 𝐴) → (𝐺𝑋) ∈ On)
201, 18, 19syl2anc 575 . . . . 5 (𝜑 → (𝐺𝑋) ∈ On)
21 omcl 7856 . . . . 5 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐺𝑋) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ∈ On)
2214, 20, 21syl2anc 575 . . . 4 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ∈ On)
23 suppssdm 7545 . . . . . . . . . . . 12 (𝐺 supp ∅) ⊆ dom 𝐺
2423, 17fssdm 6275 . . . . . . . . . . 11 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
252, 24ssexd 5007 . . . . . . . . . 10 (𝜑 → (𝐺 supp ∅) ∈ V)
26 cantnflem1.o . . . . . . . . . . . 12 𝑂 = OrdIso( E , (𝐺 supp ∅))
273, 1, 2, 26, 6cantnfcl 8814 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
2827simpld 484 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
2926oiiso 8684 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
3025, 28, 29syl2anc 575 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
31 isof1o 6800 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
3230, 31syl 17 . . . . . . . 8 (𝜑𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
33 f1ocnv 6368 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
34 f1of 6356 . . . . . . . 8 (𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂𝑂:(𝐺 supp ∅)⟶dom 𝑂)
3532, 33, 343syl 18 . . . . . . 7 (𝜑𝑂:(𝐺 supp ∅)⟶dom 𝑂)
363, 1, 2, 4, 5, 6, 7, 8cantnflem1a 8832 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
3735, 36ffvelrnd 6585 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
3827simprd 485 . . . . . 6 (𝜑 → dom 𝑂 ∈ ω)
39 elnn 7308 . . . . . 6 (((𝑂𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (𝑂𝑋) ∈ ω)
4037, 38, 39syl2anc 575 . . . . 5 (𝜑 → (𝑂𝑋) ∈ ω)
41 cantnflem1.h . . . . . . 7 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐺‘(𝑂𝑘))) +𝑜 𝑧)), ∅)
4241cantnfvalf 8812 . . . . . 6 𝐻:ω⟶On
4342ffvelrni 6583 . . . . 5 ((𝑂𝑋) ∈ ω → (𝐻‘(𝑂𝑋)) ∈ On)
4440, 43syl 17 . . . 4 (𝜑 → (𝐻‘(𝑂𝑋)) ∈ On)
45 oaword1 7872 . . . 4 ((((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ∈ On ∧ (𝐻‘(𝑂𝑋)) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ⊆ (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
4622, 44, 45syl2anc 575 . . 3 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ⊆ (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
473, 1, 2, 26, 6, 41cantnfsuc 8817 . . . . 5 ((𝜑 ∧ (𝑂𝑋) ∈ ω) → (𝐻‘suc (𝑂𝑋)) = (((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) +𝑜 (𝐻‘(𝑂𝑋))))
4840, 47mpdan 670 . . . 4 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) +𝑜 (𝐻‘(𝑂𝑋))))
49 f1ocnvfv2 6760 . . . . . . . 8 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5032, 36, 49syl2anc 575 . . . . . . 7 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5150oveq2d 6893 . . . . . 6 (𝜑 → (𝐴𝑜 (𝑂‘(𝑂𝑋))) = (𝐴𝑜 𝑋))
5250fveq2d 6415 . . . . . 6 (𝜑 → (𝐺‘(𝑂‘(𝑂𝑋))) = (𝐺𝑋))
5351, 52oveq12d 6895 . . . . 5 (𝜑 → ((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) = ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
5453oveq1d 6892 . . . 4 (𝜑 → (((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) +𝑜 (𝐻‘(𝑂𝑋))) = (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
5548, 54eqtrd 2847 . . 3 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
5646, 55sseqtr4d 3846 . 2 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ⊆ (𝐻‘suc (𝑂𝑋)))
57 onss 7223 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ⊆ On)
582, 57syl 17 . . . . . . . . . 10 (𝜑𝐵 ⊆ On)
5958sselda 3805 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
6012adantr 468 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋 ∈ On)
61 onsseleq 5984 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6259, 60, 61syl2anc 575 . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
63 orcom 888 . . . . . . . 8 ((𝑥𝑋𝑥 = 𝑋) ↔ (𝑥 = 𝑋𝑥𝑋))
6462, 63syl6bb 278 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥 = 𝑋𝑥𝑋)))
6564ifbid 4308 . . . . . 6 ((𝜑𝑥𝐵) → if(𝑥𝑋, (𝐹𝑥), ∅) = if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))
6665mpteq2dva 4945 . . . . 5 (𝜑 → (𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)))
6766fveq2d 6415 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))))
683, 1, 2cantnfs 8813 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
695, 68mpbid 223 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐵𝐴𝐹 finSupp ∅))
7069simpld 484 . . . . . . . . . 10 (𝜑𝐹:𝐵𝐴)
7170ffvelrnda 6584 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝐹𝑦) ∈ 𝐴)
7218ne0d 4130 . . . . . . . . . . 11 (𝜑𝐴 ≠ ∅)
73 on0eln0 5999 . . . . . . . . . . . 12 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
741, 73syl 17 . . . . . . . . . . 11 (𝜑 → (∅ ∈ 𝐴𝐴 ≠ ∅))
7572, 74mpbird 248 . . . . . . . . . 10 (𝜑 → ∅ ∈ 𝐴)
7675adantr 468 . . . . . . . . 9 ((𝜑𝑦𝐵) → ∅ ∈ 𝐴)
7771, 76ifcld 4331 . . . . . . . 8 ((𝜑𝑦𝐵) → if(𝑦𝑋, (𝐹𝑦), ∅) ∈ 𝐴)
7877fmpttd 6610 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴)
79 0ex 4991 . . . . . . . . 9 ∅ ∈ V
8079a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ V)
8169simprd 485 . . . . . . . 8 (𝜑𝐹 finSupp ∅)
8270, 2, 80, 81fsuppmptif 8547 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)
833, 1, 2cantnfs 8813 . . . . . . 7 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆 ↔ ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴 ∧ (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)))
8478, 82, 83mpbir2and 695 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆)
8570, 10ffvelrnd 6585 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ 𝐴)
86 eldifn 3939 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑋) → ¬ 𝑦𝑋)
8786adantl 469 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐵𝑋)) → ¬ 𝑦𝑋)
8887iffalsed 4297 . . . . . . 7 ((𝜑𝑦 ∈ (𝐵𝑋)) → if(𝑦𝑋, (𝐹𝑦), ∅) = ∅)
8988, 2suppss2 7567 . . . . . 6 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) supp ∅) ⊆ 𝑋)
90 fveq2 6411 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9190adantl 469 . . . . . . . . . 10 ((𝑥𝐵𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
9291ifeq1da 4316 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
93 eleq1w 2875 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝑋𝑥𝑋))
94 fveq2 6411 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
9593, 94ifbieq1d 4309 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑦𝑋, (𝐹𝑦), ∅) = if(𝑥𝑋, (𝐹𝑥), ∅))
96 eqid 2813 . . . . . . . . . . 11 (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) = (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))
97 fvex 6424 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
9897, 79ifex 4334 . . . . . . . . . . 11 if(𝑥𝑋, (𝐹𝑥), ∅) ∈ V
9995, 96, 98fvmpt 6506 . . . . . . . . . 10 (𝑥𝐵 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥) = if(𝑥𝑋, (𝐹𝑥), ∅))
10099ifeq2d 4305 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
10192, 100eqtr3d 2849 . . . . . . . 8 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
102 ifor 4338 . . . . . . . 8 if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅))
103101, 102syl6reqr 2866 . . . . . . 7 (𝑥𝐵 → if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
104103mpteq2ia 4941 . . . . . 6 (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
1053, 1, 2, 84, 10, 85, 89, 104cantnfp1 8828 . . . . 5 (𝜑 → ((𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))))))
106105simprd 485 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
10767, 106eqtrd 2847 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
108 onelon 5968 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐹𝑋) ∈ 𝐴) → (𝐹𝑋) ∈ On)
1091, 85, 108syl2anc 575 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ On)
110 omsuc 7846 . . . . . 6 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)))
11114, 109, 110syl2anc 575 . . . . 5 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)))
112 eloni 5953 . . . . . . . 8 ((𝐺𝑋) ∈ On → Ord (𝐺𝑋))
11320, 112syl 17 . . . . . . 7 (𝜑 → Ord (𝐺𝑋))
1149simp2d 1166 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ (𝐺𝑋))
115 ordsucss 7251 . . . . . . 7 (Ord (𝐺𝑋) → ((𝐹𝑋) ∈ (𝐺𝑋) → suc (𝐹𝑋) ⊆ (𝐺𝑋)))
116113, 114, 115sylc 65 . . . . . 6 (𝜑 → suc (𝐹𝑋) ⊆ (𝐺𝑋))
117 suceloni 7246 . . . . . . . 8 ((𝐹𝑋) ∈ On → suc (𝐹𝑋) ∈ On)
118109, 117syl 17 . . . . . . 7 (𝜑 → suc (𝐹𝑋) ∈ On)
119 omwordi 7891 . . . . . . 7 ((suc (𝐹𝑋) ∈ On ∧ (𝐺𝑋) ∈ On ∧ (𝐴𝑜 𝑋) ∈ On) → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋))))
120118, 20, 14, 119syl3anc 1483 . . . . . 6 (𝜑 → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋))))
121116, 120mpd 15 . . . . 5 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
122111, 121eqsstr3d 3844 . . . 4 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
1233, 1, 2, 84, 75, 12, 89cantnflt2 8820 . . . . 5 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴𝑜 𝑋))
124 onelon 5968 . . . . . . 7 (((𝐴𝑜 𝑋) ∈ On ∧ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴𝑜 𝑋)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
12514, 123, 124syl2anc 575 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
126 omcl 7856 . . . . . . 7 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) ∈ On)
12714, 109, 126syl2anc 575 . . . . . 6 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) ∈ On)
128 oaord 7867 . . . . . 6 ((((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On ∧ (𝐴𝑜 𝑋) ∈ On ∧ ((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) ∈ On) → (((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴𝑜 𝑋) ↔ (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋))))
129125, 14, 127, 128syl3anc 1483 . . . . 5 (𝜑 → (((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴𝑜 𝑋) ↔ (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋))))
130123, 129mpbid 223 . . . 4 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)))
131122, 130sseldd 3806 . . 3 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
132107, 131eqeltrd 2892 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
13356, 132sseldd 3806 1 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ (𝐻‘suc (𝑂𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2157  wne 2985  wral 3103  wrex 3104  {crab 3107  Vcvv 3398  cdif 3773  wss 3776  c0 4123  ifcif 4286   cuni 4637   class class class wbr 4851  {copab 4913  cmpt 4930   E cep 5230   We wwe 5276  ccnv 5317  dom cdm 5318  Ord word 5942  Oncon0 5943  suc csuc 5945  wf 6100  1-1-ontowf1o 6103  cfv 6104   Isom wiso 6105  (class class class)co 6877  cmpt2 6879  ωcom 7298   supp csupp 7532  seq𝜔cseqom 7781   +𝑜 coa 7796   ·𝑜 comu 7797  𝑜 coe 7798   finSupp cfsupp 8517  OrdIsocoi 8656   CNF ccnf 8808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-supp 7533  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-seqom 7782  df-1o 7799  df-2o 7800  df-oadd 7803  df-omul 7804  df-oexp 7805  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fsupp 8518  df-oi 8657  df-cnf 8809
This theorem is referenced by:  cantnflem1  8836
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