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Theorem cantnflem1d 8623
 Description: Lemma for cantnf 8628. (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 8619 . . . . . . . 8 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
109simp1d 1093 . . . . . . 7 (𝜑𝑋𝐵)
11 onelon 5786 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
122, 10, 11syl2anc 694 . . . . . 6 (𝜑𝑋 ∈ On)
13 oecl 7662 . . . . . 6 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
141, 12, 13syl2anc 694 . . . . 5 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
153, 1, 2cantnfs 8601 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
166, 15mpbid 222 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 474 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
1817, 10ffvelrnd 6400 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
19 onelon 5786 . . . . . 6 ((𝐴 ∈ On ∧ (𝐺𝑋) ∈ 𝐴) → (𝐺𝑋) ∈ On)
201, 18, 19syl2anc 694 . . . . 5 (𝜑 → (𝐺𝑋) ∈ On)
21 omcl 7661 . . . . 5 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐺𝑋) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ∈ On)
2214, 20, 21syl2anc 694 . . . 4 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ∈ On)
23 suppssdm 7353 . . . . . . . . . . . 12 (𝐺 supp ∅) ⊆ dom 𝐺
24 fdm 6089 . . . . . . . . . . . . 13 (𝐺:𝐵𝐴 → dom 𝐺 = 𝐵)
2517, 24syl 17 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = 𝐵)
2623, 25syl5sseq 3686 . . . . . . . . . . 11 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
272, 26ssexd 4838 . . . . . . . . . 10 (𝜑 → (𝐺 supp ∅) ∈ V)
28 cantnflem1.o . . . . . . . . . . . 12 𝑂 = OrdIso( E , (𝐺 supp ∅))
293, 1, 2, 28, 6cantnfcl 8602 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
3029simpld 474 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
3128oiiso 8483 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
3227, 30, 31syl2anc 694 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
33 isof1o 6613 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
3432, 33syl 17 . . . . . . . 8 (𝜑𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
35 f1ocnv 6187 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
36 f1of 6175 . . . . . . . 8 (𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂𝑂:(𝐺 supp ∅)⟶dom 𝑂)
3734, 35, 363syl 18 . . . . . . 7 (𝜑𝑂:(𝐺 supp ∅)⟶dom 𝑂)
383, 1, 2, 4, 5, 6, 7, 8cantnflem1a 8620 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
3937, 38ffvelrnd 6400 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
4029simprd 478 . . . . . 6 (𝜑 → dom 𝑂 ∈ ω)
41 elnn 7117 . . . . . 6 (((𝑂𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (𝑂𝑋) ∈ ω)
4239, 40, 41syl2anc 694 . . . . 5 (𝜑 → (𝑂𝑋) ∈ ω)
43 cantnflem1.h . . . . . . 7 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐺‘(𝑂𝑘))) +𝑜 𝑧)), ∅)
4443cantnfvalf 8600 . . . . . 6 𝐻:ω⟶On
4544ffvelrni 6398 . . . . 5 ((𝑂𝑋) ∈ ω → (𝐻‘(𝑂𝑋)) ∈ On)
4642, 45syl 17 . . . 4 (𝜑 → (𝐻‘(𝑂𝑋)) ∈ On)
47 oaword1 7677 . . . 4 ((((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ∈ On ∧ (𝐻‘(𝑂𝑋)) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ⊆ (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
4822, 46, 47syl2anc 694 . . 3 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ⊆ (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
493, 1, 2, 28, 6, 43cantnfsuc 8605 . . . . 5 ((𝜑 ∧ (𝑂𝑋) ∈ ω) → (𝐻‘suc (𝑂𝑋)) = (((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) +𝑜 (𝐻‘(𝑂𝑋))))
5042, 49mpdan 703 . . . 4 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) +𝑜 (𝐻‘(𝑂𝑋))))
51 f1ocnvfv2 6573 . . . . . . . 8 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5234, 38, 51syl2anc 694 . . . . . . 7 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5352oveq2d 6706 . . . . . 6 (𝜑 → (𝐴𝑜 (𝑂‘(𝑂𝑋))) = (𝐴𝑜 𝑋))
5452fveq2d 6233 . . . . . 6 (𝜑 → (𝐺‘(𝑂‘(𝑂𝑋))) = (𝐺𝑋))
5553, 54oveq12d 6708 . . . . 5 (𝜑 → ((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) = ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
5655oveq1d 6705 . . . 4 (𝜑 → (((𝐴𝑜 (𝑂‘(𝑂𝑋))) ·𝑜 (𝐺‘(𝑂‘(𝑂𝑋)))) +𝑜 (𝐻‘(𝑂𝑋))) = (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
5750, 56eqtrd 2685 . . 3 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) +𝑜 (𝐻‘(𝑂𝑋))))
5848, 57sseqtr4d 3675 . 2 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)) ⊆ (𝐻‘suc (𝑂𝑋)))
59 onss 7032 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ⊆ On)
602, 59syl 17 . . . . . . . . . 10 (𝜑𝐵 ⊆ On)
6160sselda 3636 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
6212adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋 ∈ On)
63 onsseleq 5803 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6461, 62, 63syl2anc 694 . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
65 orcom 401 . . . . . . . 8 ((𝑥𝑋𝑥 = 𝑋) ↔ (𝑥 = 𝑋𝑥𝑋))
6664, 65syl6bb 276 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥 = 𝑋𝑥𝑋)))
6766ifbid 4141 . . . . . 6 ((𝜑𝑥𝐵) → if(𝑥𝑋, (𝐹𝑥), ∅) = if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))
6867mpteq2dva 4777 . . . . 5 (𝜑 → (𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)))
6968fveq2d 6233 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))))
703, 1, 2cantnfs 8601 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
715, 70mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐵𝐴𝐹 finSupp ∅))
7271simpld 474 . . . . . . . . . 10 (𝜑𝐹:𝐵𝐴)
7372ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝐹𝑦) ∈ 𝐴)
74 ne0i 3954 . . . . . . . . . . . 12 ((𝐺𝑋) ∈ 𝐴𝐴 ≠ ∅)
7518, 74syl 17 . . . . . . . . . . 11 (𝜑𝐴 ≠ ∅)
76 on0eln0 5818 . . . . . . . . . . . 12 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
771, 76syl 17 . . . . . . . . . . 11 (𝜑 → (∅ ∈ 𝐴𝐴 ≠ ∅))
7875, 77mpbird 247 . . . . . . . . . 10 (𝜑 → ∅ ∈ 𝐴)
7978adantr 480 . . . . . . . . 9 ((𝜑𝑦𝐵) → ∅ ∈ 𝐴)
8073, 79ifcld 4164 . . . . . . . 8 ((𝜑𝑦𝐵) → if(𝑦𝑋, (𝐹𝑦), ∅) ∈ 𝐴)
81 eqid 2651 . . . . . . . 8 (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) = (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))
8280, 81fmptd 6425 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴)
83 0ex 4823 . . . . . . . . 9 ∅ ∈ V
8483a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ V)
8571simprd 478 . . . . . . . 8 (𝜑𝐹 finSupp ∅)
8672, 2, 84, 85fsuppmptif 8346 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)
873, 1, 2cantnfs 8601 . . . . . . 7 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆 ↔ ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴 ∧ (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)))
8882, 86, 87mpbir2and 977 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆)
8972, 10ffvelrnd 6400 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ 𝐴)
90 eldifn 3766 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑋) → ¬ 𝑦𝑋)
9190adantl 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐵𝑋)) → ¬ 𝑦𝑋)
9291iffalsed 4130 . . . . . . 7 ((𝜑𝑦 ∈ (𝐵𝑋)) → if(𝑦𝑋, (𝐹𝑦), ∅) = ∅)
9392, 2suppss2 7374 . . . . . 6 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) supp ∅) ⊆ 𝑋)
94 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9594adantl 481 . . . . . . . . . 10 ((𝑥𝐵𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
9695ifeq1da 4149 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
97 eleq1 2718 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝑋𝑥𝑋))
98 fveq2 6229 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
9997, 98ifbieq1d 4142 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑦𝑋, (𝐹𝑦), ∅) = if(𝑥𝑋, (𝐹𝑥), ∅))
100 fvex 6239 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
101100, 83ifex 4189 . . . . . . . . . . 11 if(𝑥𝑋, (𝐹𝑥), ∅) ∈ V
10299, 81, 101fvmpt 6321 . . . . . . . . . 10 (𝑥𝐵 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥) = if(𝑥𝑋, (𝐹𝑥), ∅))
103102ifeq2d 4138 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
10496, 103eqtr3d 2687 . . . . . . . 8 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
105 ifor 4168 . . . . . . . 8 if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅))
106104, 105syl6reqr 2704 . . . . . . 7 (𝑥𝐵 → if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
107106mpteq2ia 4773 . . . . . 6 (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
1083, 1, 2, 88, 10, 89, 93, 107cantnfp1 8616 . . . . 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), ∅))))))
109108simprd 478 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
11069, 109eqtrd 2685 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
111 onelon 5786 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐹𝑋) ∈ 𝐴) → (𝐹𝑋) ∈ On)
1121, 89, 111syl2anc 694 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ On)
113 omsuc 7651 . . . . . 6 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)))
11414, 112, 113syl2anc 694 . . . . 5 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) = (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)))
115 eloni 5771 . . . . . . . 8 ((𝐺𝑋) ∈ On → Ord (𝐺𝑋))
11620, 115syl 17 . . . . . . 7 (𝜑 → Ord (𝐺𝑋))
1179simp2d 1094 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ (𝐺𝑋))
118 ordsucss 7060 . . . . . . 7 (Ord (𝐺𝑋) → ((𝐹𝑋) ∈ (𝐺𝑋) → suc (𝐹𝑋) ⊆ (𝐺𝑋)))
119116, 117, 118sylc 65 . . . . . 6 (𝜑 → suc (𝐹𝑋) ⊆ (𝐺𝑋))
120 suceloni 7055 . . . . . . . 8 ((𝐹𝑋) ∈ On → suc (𝐹𝑋) ∈ On)
121112, 120syl 17 . . . . . . 7 (𝜑 → suc (𝐹𝑋) ∈ On)
122 omwordi 7696 . . . . . . 7 ((suc (𝐹𝑋) ∈ On ∧ (𝐺𝑋) ∈ On ∧ (𝐴𝑜 𝑋) ∈ On) → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋))))
123121, 20, 14, 122syl3anc 1366 . . . . . 6 (𝜑 → (suc (𝐹𝑋) ⊆ (𝐺𝑋) → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋))))
124119, 123mpd 15 . . . . 5 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 suc (𝐹𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
125114, 124eqsstr3d 3673 . . . 4 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)) ⊆ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
1263, 1, 2, 88, 78, 12, 93cantnflt2 8608 . . . . 5 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴𝑜 𝑋))
127 onelon 5786 . . . . . . 7 (((𝐴𝑜 𝑋) ∈ On ∧ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴𝑜 𝑋)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
12814, 126, 127syl2anc 694 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
129 omcl 7661 . . . . . . 7 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) ∈ On)
13014, 112, 129syl2anc 694 . . . . . 6 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) ∈ On)
131 oaord 7672 . . . . . 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))))
132128, 14, 130, 131syl3anc 1366 . . . . 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))))
133126, 132mpbid 222 . . . 4 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 (𝐴𝑜 𝑋)))
134125, 133sseldd 3637 . . 3 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 (𝐹𝑋)) +𝑜 ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
135110, 134eqeltrd 2730 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ ((𝐴𝑜 𝑋) ·𝑜 (𝐺𝑋)))
13658, 135sseldd 3637 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 196   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ∅c0 3948  ifcif 4119  ∪ cuni 4468   class class class wbr 4685  {copab 4745   ↦ cmpt 4762   E cep 5057   We wwe 5101  ◡ccnv 5142  dom cdm 5143  Ord word 5760  Oncon0 5761  suc csuc 5763  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926   Isom wiso 5927  (class class class)co 6690   ↦ cmpt2 6692  ωcom 7107   supp csupp 7340  seq𝜔cseqom 7587   +𝑜 coa 7602   ·𝑜 comu 7603   ↑𝑜 coe 7604   finSupp cfsupp 8316  OrdIsocoi 8455   CNF ccnf 8596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-cnf 8597 This theorem is referenced by:  cantnflem1  8624
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