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Theorem cantnflem1d 9729
Description: Lemma for cantnf 9734. (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 9725 . . . . . . . 8 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
109simp1d 1142 . . . . . . 7 (𝜑𝑋𝐵)
11 onelon 6408 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
122, 10, 11syl2anc 584 . . . . . 6 (𝜑𝑋 ∈ On)
13 oecl 8576 . . . . . 6 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
141, 12, 13syl2anc 584 . . . . 5 (𝜑 → (𝐴o 𝑋) ∈ On)
153, 1, 2cantnfs 9707 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
166, 15mpbid 232 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 494 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
1817, 10ffvelcdmd 7104 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
19 onelon 6408 . . . . . 6 ((𝐴 ∈ On ∧ (𝐺𝑋) ∈ 𝐴) → (𝐺𝑋) ∈ On)
201, 18, 19syl2anc 584 . . . . 5 (𝜑 → (𝐺𝑋) ∈ On)
21 omcl 8575 . . . . 5 (((𝐴o 𝑋) ∈ On ∧ (𝐺𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
2214, 20, 21syl2anc 584 . . . 4 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On)
23 ovexd 7467 . . . . . . . . . 10 (𝜑 → (𝐺 supp ∅) ∈ V)
24 cantnflem1.o . . . . . . . . . . . 12 𝑂 = OrdIso( E , (𝐺 supp ∅))
253, 1, 2, 24, 6cantnfcl 9708 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
2625simpld 494 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
2724oiiso 9578 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
2823, 26, 27syl2anc 584 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
29 isof1o 7344 . . . . . . . . 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 6859 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
32 f1of 6847 . . . . . . . 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 9726 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
3533, 34ffvelcdmd 7104 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
3625simprd 495 . . . . . 6 (𝜑 → dom 𝑂 ∈ ω)
37 elnn 7899 . . . . . 6 (((𝑂𝑋) ∈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (𝑂𝑋) ∈ ω)
3835, 36, 37syl2anc 584 . . . . 5 (𝜑 → (𝑂𝑋) ∈ ω)
39 cantnflem1.h . . . . . . 7 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑂𝑘)) ·o (𝐺‘(𝑂𝑘))) +o 𝑧)), ∅)
4039cantnfvalf 9706 . . . . . 6 𝐻:ω⟶On
4140ffvelcdmi 7102 . . . . 5 ((𝑂𝑋) ∈ ω → (𝐻‘(𝑂𝑋)) ∈ On)
4238, 41syl 17 . . . 4 (𝜑 → (𝐻‘(𝑂𝑋)) ∈ On)
43 oaword1 8591 . . . 4 ((((𝐴o 𝑋) ·o (𝐺𝑋)) ∈ On ∧ (𝐻‘(𝑂𝑋)) ∈ On) → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
4422, 42, 43syl2anc 584 . . 3 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
453, 1, 2, 24, 6, 39cantnfsuc 9711 . . . . 5 ((𝜑 ∧ (𝑂𝑋) ∈ ω) → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
4638, 45mpdan 687 . . . 4 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))))
47 f1ocnvfv2 7298 . . . . . . . 8 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
4830, 34, 47syl2anc 584 . . . . . . 7 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
4948oveq2d 7448 . . . . . 6 (𝜑 → (𝐴o (𝑂‘(𝑂𝑋))) = (𝐴o 𝑋))
5048fveq2d 6909 . . . . . 6 (𝜑 → (𝐺‘(𝑂‘(𝑂𝑋))) = (𝐺𝑋))
5149, 50oveq12d 7450 . . . . 5 (𝜑 → ((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) = ((𝐴o 𝑋) ·o (𝐺𝑋)))
5251oveq1d 7447 . . . 4 (𝜑 → (((𝐴o (𝑂‘(𝑂𝑋))) ·o (𝐺‘(𝑂‘(𝑂𝑋)))) +o (𝐻‘(𝑂𝑋))) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5346, 52eqtrd 2776 . . 3 (𝜑 → (𝐻‘suc (𝑂𝑋)) = (((𝐴o 𝑋) ·o (𝐺𝑋)) +o (𝐻‘(𝑂𝑋))))
5444, 53sseqtrrd 4020 . 2 (𝜑 → ((𝐴o 𝑋) ·o (𝐺𝑋)) ⊆ (𝐻‘suc (𝑂𝑋)))
55 onss 7806 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ⊆ On)
562, 55syl 17 . . . . . . . . . 10 (𝜑𝐵 ⊆ On)
5756sselda 3982 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑥 ∈ On)
5812adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋 ∈ On)
59 onsseleq 6424 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6057, 58, 59syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
61 orcom 870 . . . . . . . 8 ((𝑥𝑋𝑥 = 𝑋) ↔ (𝑥 = 𝑋𝑥𝑋))
6260, 61bitrdi 287 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥𝑋 ↔ (𝑥 = 𝑋𝑥𝑋)))
6362ifbid 4548 . . . . . 6 ((𝜑𝑥𝐵) → if(𝑥𝑋, (𝐹𝑥), ∅) = if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))
6463mpteq2dva 5241 . . . . 5 (𝜑 → (𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)))
6564fveq2d 6909 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))))
663, 1, 2cantnfs 9707 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
675, 66mpbid 232 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐵𝐴𝐹 finSupp ∅))
6867simpld 494 . . . . . . . . . 10 (𝜑𝐹:𝐵𝐴)
6968ffvelcdmda 7103 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝐹𝑦) ∈ 𝐴)
7018ne0d 4341 . . . . . . . . . . 11 (𝜑𝐴 ≠ ∅)
71 on0eln0 6439 . . . . . . . . . . . 12 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
721, 71syl 17 . . . . . . . . . . 11 (𝜑 → (∅ ∈ 𝐴𝐴 ≠ ∅))
7370, 72mpbird 257 . . . . . . . . . 10 (𝜑 → ∅ ∈ 𝐴)
7473adantr 480 . . . . . . . . 9 ((𝜑𝑦𝐵) → ∅ ∈ 𝐴)
7569, 74ifcld 4571 . . . . . . . 8 ((𝜑𝑦𝐵) → if(𝑦𝑋, (𝐹𝑦), ∅) ∈ 𝐴)
7675fmpttd 7134 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴)
77 0ex 5306 . . . . . . . . 9 ∅ ∈ V
7877a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ V)
7967simprd 495 . . . . . . . 8 (𝜑𝐹 finSupp ∅)
8068, 2, 78, 79fsuppmptif 9440 . . . . . . 7 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)
813, 1, 2cantnfs 9707 . . . . . . 7 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆 ↔ ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)):𝐵𝐴 ∧ (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) finSupp ∅)))
8276, 80, 81mpbir2and 713 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) ∈ 𝑆)
8368, 10ffvelcdmd 7104 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ 𝐴)
84 eldifn 4131 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑋) → ¬ 𝑦𝑋)
8584adantl 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐵𝑋)) → ¬ 𝑦𝑋)
8685iffalsed 4535 . . . . . . 7 ((𝜑𝑦 ∈ (𝐵𝑋)) → if(𝑦𝑋, (𝐹𝑦), ∅) = ∅)
8786, 2suppss2 8226 . . . . . 6 (𝜑 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) supp ∅) ⊆ 𝑋)
88 ifor 4579 . . . . . . . 8 if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅))
89 fveq2 6905 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9089adantl 481 . . . . . . . . . 10 ((𝑥𝐵𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
9190ifeq1da 4556 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
92 eleq1w 2823 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝑋𝑥𝑋))
93 fveq2 6905 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
9492, 93ifbieq1d 4549 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑦𝑋, (𝐹𝑦), ∅) = if(𝑥𝑋, (𝐹𝑥), ∅))
95 eqid 2736 . . . . . . . . . . 11 (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)) = (𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))
96 fvex 6918 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
9796, 77ifex 4575 . . . . . . . . . . 11 if(𝑥𝑋, (𝐹𝑥), ∅) ∈ V
9894, 95, 97fvmpt 7015 . . . . . . . . . 10 (𝑥𝐵 → ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥) = if(𝑥𝑋, (𝐹𝑥), ∅))
9998ifeq2d 4545 . . . . . . . . 9 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑥), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
10091, 99eqtr3d 2778 . . . . . . . 8 (𝑥𝐵 → if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)) = if(𝑥 = 𝑋, (𝐹𝑥), if(𝑥𝑋, (𝐹𝑥), ∅)))
10188, 100eqtr4id 2795 . . . . . . 7 (𝑥𝐵 → if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅) = if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
102101mpteq2ia 5244 . . . . . 6 (𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅)) = (𝑥𝐵 ↦ if(𝑥 = 𝑋, (𝐹𝑋), ((𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))‘𝑥)))
1033, 1, 2, 82, 10, 83, 87, 102cantnfp1 9722 . . . . 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 495 . . . 4 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if((𝑥 = 𝑋𝑥𝑋), (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
10565, 104eqtrd 2776 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))))
106 onelon 6408 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐹𝑋) ∈ 𝐴) → (𝐹𝑋) ∈ On)
1071, 83, 106syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ On)
108 omsuc 8565 . . . . . 6 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
10914, 107, 108syl2anc 584 . . . . 5 (𝜑 → ((𝐴o 𝑋) ·o suc (𝐹𝑋)) = (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
110 eloni 6393 . . . . . . . 8 ((𝐺𝑋) ∈ On → Ord (𝐺𝑋))
11120, 110syl 17 . . . . . . 7 (𝜑 → Ord (𝐺𝑋))
1129simp2d 1143 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ (𝐺𝑋))
113 ordsucss 7839 . . . . . . 7 (Ord (𝐺𝑋) → ((𝐹𝑋) ∈ (𝐺𝑋) → suc (𝐹𝑋) ⊆ (𝐺𝑋)))
114111, 112, 113sylc 65 . . . . . 6 (𝜑 → suc (𝐹𝑋) ⊆ (𝐺𝑋))
115 onsuc 7832 . . . . . . . 8 ((𝐹𝑋) ∈ On → suc (𝐹𝑋) ∈ On)
116107, 115syl 17 . . . . . . 7 (𝜑 → suc (𝐹𝑋) ∈ On)
117 omwordi 8610 . . . . . . 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 1372 . . . . . 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 4018 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)) ⊆ ((𝐴o 𝑋) ·o (𝐺𝑋)))
1213, 1, 2, 82, 73, 12, 87cantnflt2 9714 . . . . 5 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋))
122 onelon 6408 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ (𝐴o 𝑋)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
12314, 121, 122syl2anc 584 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅))) ∈ On)
124 omcl 8575 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ (𝐹𝑋) ∈ On) → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
12514, 107, 124syl2anc 584 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o (𝐹𝑋)) ∈ On)
126 oaord 8586 . . . . . 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 1372 . . . . 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 232 . . . 4 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ (((𝐴o 𝑋) ·o (𝐹𝑋)) +o (𝐴o 𝑋)))
129120, 128sseldd 3983 . . 3 (𝜑 → (((𝐴o 𝑋) ·o (𝐹𝑋)) +o ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦𝑋, (𝐹𝑦), ∅)))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
130105, 129eqeltrd 2840 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ ((𝐴o 𝑋) ·o (𝐺𝑋)))
13154, 130sseldd 3983 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 206  wa 395  wo 847   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  cdif 3947  wss 3950  c0 4332  ifcif 4524   cuni 4906   class class class wbr 5142  {copab 5204  cmpt 5224   E cep 5582   We wwe 5635  ccnv 5683  dom cdm 5684  Ord word 6382  Oncon0 6383  suc csuc 6385  wf 6556  1-1-ontowf1o 6559  cfv 6560   Isom wiso 6561  (class class class)co 7432  cmpo 7434  ωcom 7888   supp csupp 8186  seqωcseqom 8488   +o coa 8504   ·o comu 8505  o coe 8506   finSupp cfsupp 9402  OrdIsocoi 9550   CNF ccnf 9702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqom 8489  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-oi 9551  df-cnf 9703
This theorem is referenced by:  cantnflem1  9730
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