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Theorem cantnflem1d 9139
 Description: Lemma for cantnf 9144. (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 9135 . . . . . . . 8 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
109simp1d 1139 . . . . . . 7 (𝜑𝑋𝐵)
11 onelon 6194 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
122, 10, 11syl2anc 587 . . . . . 6 (𝜑𝑋 ∈ On)
13 oecl 8149 . . . . . 6 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
141, 12, 13syl2anc 587 . . . . 5 (𝜑 → (𝐴o 𝑋) ∈ On)
153, 1, 2cantnfs 9117 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
166, 15mpbid 235 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 498 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
1817, 10ffvelrnd 6834 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
19 onelon 6194 . . . . . 6 ((𝐴 ∈ On ∧ (𝐺𝑋) ∈ 𝐴) → (𝐺𝑋) ∈ On)
201, 18, 19syl2anc 587 . . . . 5 (𝜑 → (𝐺𝑋) ∈ On)
21 omcl 8148 . . . . 5 (((𝐴o 𝑋) ∈ On ∧ (𝐺𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
2214, 20, 21syl2anc 587 . . . 4 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
23 ovexd 7175 . . . . . . . . . 10 (𝜑 → (𝐺 supp ∅) ∈ V)
24 cantnflem1.o . . . . . . . . . . . 12 𝑂 = OrdIso( E , (𝐺 supp ∅))
253, 1, 2, 24, 6cantnfcl 9118 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
2625simpld 498 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
2724oiiso 8989 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
2823, 26, 27syl2anc 587 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
29 isof1o 7060 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
3028, 29syl 17 . . . . . . . 8 (𝜑𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
31 f1ocnv 6609 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32 f1of 6597 . . . . . . . 8 (𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂𝑂:(𝐺 supp ∅)⟶dom 𝑂)
3330, 31, 323syl 18 . . . . . . 7 (𝜑𝑂:(𝐺 supp ∅)⟶dom 𝑂)
343, 1, 2, 4, 5, 6, 7, 8cantnflem1a 9136 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
3533, 34ffvelrnd 6834 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
3625simprd 499 . . . . . 6 (𝜑 → dom 𝑂 ∈ ω)
37 elnn 7575 . . . . . 6 (((𝑂𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (𝑂𝑋) ∈ ω)
3835, 36, 37syl2anc 587 . . . . 5 (𝜑 → (𝑂𝑋) ∈ ω)
39 cantnflem1.h . . . . . . 7 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑂𝑘)) ·o (𝐺‘(𝑂𝑘))) +o 𝑧)), ∅)
4039cantnfvalf 9116 . . . . . 6 𝐻:ω⟶On
4140ffvelrni 6832 . . . . 5 ((𝑂𝑋) ∈ ω → (𝐻‘(𝑂𝑋)) ∈ On)
4238, 41syl 17 . . . 4 (𝜑 → (𝐻‘(𝑂𝑋)) ∈ On)
43 oaword1 8165 . . . 4 ((((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On ∧ (𝐻‘(𝑂𝑋)) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
4422, 42, 43syl2anc 587 . . 3 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
453, 1, 2, 24, 6, 39cantnfsuc 9121 . . . . 5 ((𝜑 ∧ (𝑂𝑋) ∈ ω) → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
4638, 45mpdan 686 . . . 4 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
47 f1ocnvfv2 7017 . . . . . . . 8 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
4830, 34, 47syl2anc 587 . . . . . . 7 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
4948oveq2d 7156 . . . . . 6 (𝜑 → (𝐴o (𝑂‘(𝑂𝑋))) = (𝐴o 𝑋))
5048fveq2d 6656 . . . . . 6 (𝜑 → (𝐺‘(𝑂‘(𝑂𝑋))) = (𝐺𝑋))
5149, 50oveq12d 7158 . . . . 5 (𝜑 → ((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) = ((𝐴o 𝑋) ·o (𝐺𝑋)))
5251oveq1d 7155 . . . 4 (𝜑 → (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5346, 52eqtrd 2857 . . 3 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5444, 53sseqtrrd 3983 . 2 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (𝐻‘suc (𝑂𝑋)))
55 onss 7490 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ⊆ On)
562, 55syl 17 . . . . . . . . . 10 (𝜑𝐵 ⊆ On)
5756sselda 3942 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
5812adantr 484 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋 ∈ On)
59 onsseleq 6210 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6057, 58, 59syl2anc 587 . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
61 orcom 867 . . . . . . . 8 ((𝑥𝑋𝑥 = 𝑋) ↔ (𝑥 = 𝑋𝑥𝑋))
6260, 61syl6bb 290 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥 = 𝑋𝑥𝑋)))
6362ifbid 4461 . . . . . 6 ((𝜑𝑥𝐵) → if(𝑥𝑋, (𝐹𝑥), ∅) = if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))
6463mpteq2dva 5137 . . . . 5 (𝜑 → (𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)))
6564fveq2d 6656 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))))
663, 1, 2cantnfs 9117 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
675, 66mpbid 235 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐵𝐴𝐹 finSupp ∅))
6867simpld 498 . . . . . . . . . 10 (𝜑𝐹:𝐵𝐴)
6968ffvelrnda 6833 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝐹𝑦) ∈ 𝐴)
7018ne0d 4273 . . . . . . . . . . 11 (𝜑𝐴 ≠ ∅)
71 on0eln0 6224 . . . . . . . . . . . 12 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
721, 71syl 17 . . . . . . . . . . 11 (𝜑 → (∅ ∈ 𝐴𝐴 ≠ ∅))
7370, 72mpbird 260 . . . . . . . . . 10 (𝜑 → ∅ ∈ 𝐴)
7473adantr 484 . . . . . . . . 9 ((𝜑𝑦𝐵) → ∅ ∈ 𝐴)
7569, 74ifcld 4484 . . . . . . . 8 ((𝜑𝑦𝐵) → if(𝑦𝑋, (𝐹𝑦), ∅) ∈ 𝐴)
7675fmpttd 6861 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴)
77 0ex 5187 . . . . . . . . 9 ∅ ∈ V
7877a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ V)
7967simprd 499 . . . . . . . 8 (𝜑𝐹 finSupp ∅)
8068, 2, 78, 79fsuppmptif 8851 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)
813, 1, 2cantnfs 9117 . . . . . . 7 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆 ↔ ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴 ∧ (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)))
8276, 80, 81mpbir2and 712 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆)
8368, 10ffvelrnd 6834 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ 𝐴)
84 eldifn 4079 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑋) → ¬ 𝑦𝑋)
8584adantl 485 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐵𝑋)) → ¬ 𝑦𝑋)
8685iffalsed 4450 . . . . . . 7 ((𝜑𝑦 ∈ (𝐵𝑋)) → if(𝑦𝑋, (𝐹𝑦), ∅) = ∅)
8786, 2suppss2 7851 . . . . . 6 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) supp ∅) ⊆ 𝑋)
88 fveq2 6652 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
8988adantl 485 . . . . . . . . . 10 ((𝑥𝐵𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
9089ifeq1da 4469 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
91 eleq1w 2896 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝑋𝑥𝑋))
92 fveq2 6652 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
9391, 92ifbieq1d 4462 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑦𝑋, (𝐹𝑦), ∅) = if(𝑥𝑋, (𝐹𝑥), ∅))
94 eqid 2822 . . . . . . . . . . 11 (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) = (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))
95 fvex 6665 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
9695, 77ifex 4487 . . . . . . . . . . 11 if(𝑥𝑋, (𝐹𝑥), ∅) ∈ V
9793, 94, 96fvmpt 6750 . . . . . . . . . 10 (𝑥𝐵 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥) = if(𝑥𝑋, (𝐹𝑥), ∅))
9897ifeq2d 4458 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
9990, 98eqtr3d 2859 . . . . . . . 8 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
100 ifor 4491 . . . . . . . 8 if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅))
10199, 100syl6reqr 2876 . . . . . . 7 (𝑥𝐵 → if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
102101mpteq2ia 5133 . . . . . 6 (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
1033, 1, 2, 82, 10, 83, 87, 102cantnfp1 9132 . . . . 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), ∅))))))
104103simprd 499 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
10565, 104eqtrd 2857 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
106 onelon 6194 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐹𝑋) ∈ 𝐴) → (𝐹𝑋) ∈ On)
1071, 83, 106syl2anc 587 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ On)
108 omsuc 8138 . . . . . 6 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
10914, 107, 108syl2anc 587 . . . . 5 (𝜑 → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
110 eloni 6179 . . . . . . . 8 ((𝐺𝑋) ∈ On → Ord (𝐺𝑋))
11120, 110syl 17 . . . . . . 7 (𝜑 → Ord (𝐺𝑋))
1129simp2d 1140 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ (𝐺𝑋))
113 ordsucss 7518 . . . . . . 7 (Ord (𝐺𝑋) → ((𝐹𝑋) ∈ (𝐺𝑋) → suc (𝐹𝑋) ⊆ (𝐺𝑋)))
114111, 112, 113sylc 65 . . . . . 6 (𝜑 → suc (𝐹𝑋) ⊆ (𝐺𝑋))
115 suceloni 7513 . . . . . . . 8 ((𝐹𝑋) ∈ On → suc (𝐹𝑋) ∈ On)
116107, 115syl 17 . . . . . . 7 (𝜑 → suc (𝐹𝑋) ∈ On)
117 omwordi 8184 . . . . . . 7 ((suc (𝐹𝑋) ∈ On ∧ (𝐺𝑋) ∈ On ∧ (𝐴o 𝑋) ∈ On) → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋))))
118116, 20, 14, 117syl3anc 1368 . . . . . 6 (𝜑 → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋))))
119114, 118mpd 15 . . . . 5 (𝜑 → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋)))
120109, 119eqsstrrd 3981 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋)))
1213, 1, 2, 82, 73, 12, 87cantnflt2 9124 . . . . 5 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋))
122 onelon 6194 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
12314, 121, 122syl2anc 587 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
124 omcl 8148 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
12514, 107, 124syl2anc 587 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
126 oaord 8160 . . . . . 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))))
127123, 14, 125, 126syl3anc 1368 . . . . 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))))
128121, 127mpbid 235 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
129120, 128sseldd 3943 . . 3 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
130105, 129eqeltrd 2914 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
13154, 130sseldd 3943 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 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∀wral 3130  ∃wrex 3131  {crab 3134  Vcvv 3469   ∖ cdif 3905   ⊆ wss 3908  ∅c0 4265  ifcif 4439  ∪ cuni 4813   class class class wbr 5042  {copab 5104   ↦ cmpt 5122   E cep 5441   We wwe 5490  ◡ccnv 5531  dom cdm 5532  Ord word 6168  Oncon0 6169  suc csuc 6171  ⟶wf 6330  –1-1-onto→wf1o 6333  ‘cfv 6334   Isom wiso 6335  (class class class)co 7140   ∈ cmpo 7142  ωcom 7565   supp csupp 7817  seqωcseqom 8070   +o coa 8086   ·o comu 8087   ↑o coe 8088   finSupp cfsupp 8821  OrdIsocoi 8961   CNF ccnf 9112 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-cnf 9113 This theorem is referenced by:  cantnflem1  9140
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