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Theorem cantnflt 9671
Description: An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴 ↑o 𝐢 where 𝐢 is larger than any exponent (πΊβ€˜π‘₯), π‘₯ ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐡)
cantnfs.a (πœ‘ β†’ 𝐴 ∈ On)
cantnfs.b (πœ‘ β†’ 𝐡 ∈ On)
cantnfcl.g 𝐺 = OrdIso( E , (𝐹 supp βˆ…))
cantnfcl.f (πœ‘ β†’ 𝐹 ∈ 𝑆)
cantnfval.h 𝐻 = seqΟ‰((π‘˜ ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (πΊβ€˜π‘˜)) Β·o (πΉβ€˜(πΊβ€˜π‘˜))) +o 𝑧)), βˆ…)
cantnflt.a (πœ‘ β†’ βˆ… ∈ 𝐴)
cantnflt.k (πœ‘ β†’ 𝐾 ∈ suc dom 𝐺)
cantnflt.c (πœ‘ β†’ 𝐢 ∈ On)
cantnflt.s (πœ‘ β†’ (𝐺 β€œ 𝐾) βŠ† 𝐢)
Assertion
Ref Expression
cantnflt (πœ‘ β†’ (π»β€˜πΎ) ∈ (𝐴 ↑o 𝐢))
Distinct variable groups:   𝑧,π‘˜,𝐡   𝑧,𝐢   𝐴,π‘˜,𝑧   π‘˜,𝐹,𝑧   𝑆,π‘˜,𝑧   π‘˜,𝐺,𝑧   π‘˜,𝐾,𝑧   πœ‘,π‘˜,𝑧
Allowed substitution hints:   𝐢(π‘˜)   𝐻(𝑧,π‘˜)

Proof of Theorem cantnflt
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfs.a . . . 4 (πœ‘ β†’ 𝐴 ∈ On)
2 cantnflt.c . . . 4 (πœ‘ β†’ 𝐢 ∈ On)
3 cantnflt.a . . . 4 (πœ‘ β†’ βˆ… ∈ 𝐴)
4 oen0 8590 . . . 4 (((𝐴 ∈ On ∧ 𝐢 ∈ On) ∧ βˆ… ∈ 𝐴) β†’ βˆ… ∈ (𝐴 ↑o 𝐢))
51, 2, 3, 4syl21anc 835 . . 3 (πœ‘ β†’ βˆ… ∈ (𝐴 ↑o 𝐢))
6 fveq2 6891 . . . . 5 (𝐾 = βˆ… β†’ (π»β€˜πΎ) = (π»β€˜βˆ…))
7 0ex 5307 . . . . . 6 βˆ… ∈ V
8 cantnfval.h . . . . . . 7 𝐻 = seqΟ‰((π‘˜ ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (πΊβ€˜π‘˜)) Β·o (πΉβ€˜(πΊβ€˜π‘˜))) +o 𝑧)), βˆ…)
98seqom0g 8460 . . . . . 6 (βˆ… ∈ V β†’ (π»β€˜βˆ…) = βˆ…)
107, 9ax-mp 5 . . . . 5 (π»β€˜βˆ…) = βˆ…
116, 10eqtrdi 2787 . . . 4 (𝐾 = βˆ… β†’ (π»β€˜πΎ) = βˆ…)
1211eleq1d 2817 . . 3 (𝐾 = βˆ… β†’ ((π»β€˜πΎ) ∈ (𝐴 ↑o 𝐢) ↔ βˆ… ∈ (𝐴 ↑o 𝐢)))
135, 12syl5ibrcom 246 . 2 (πœ‘ β†’ (𝐾 = βˆ… β†’ (π»β€˜πΎ) ∈ (𝐴 ↑o 𝐢)))
142adantr 480 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ 𝐢 ∈ On)
15 eloni 6374 . . . . . . 7 (𝐢 ∈ On β†’ Ord 𝐢)
1614, 15syl 17 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ Ord 𝐢)
17 cantnflt.s . . . . . . . 8 (πœ‘ β†’ (𝐺 β€œ 𝐾) βŠ† 𝐢)
1817adantr 480 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (𝐺 β€œ 𝐾) βŠ† 𝐢)
19 cantnfcl.g . . . . . . . . . 10 𝐺 = OrdIso( E , (𝐹 supp βˆ…))
2019oif 9529 . . . . . . . . 9 𝐺:dom 𝐺⟢(𝐹 supp βˆ…)
21 ffn 6717 . . . . . . . . 9 (𝐺:dom 𝐺⟢(𝐹 supp βˆ…) β†’ 𝐺 Fn dom 𝐺)
2220, 21mp1i 13 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ 𝐺 Fn dom 𝐺)
23 cantnflt.k . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ suc dom 𝐺)
2419oicl 9528 . . . . . . . . . . . . 13 Ord dom 𝐺
25 ordsuc 7805 . . . . . . . . . . . . 13 (Ord dom 𝐺 ↔ Ord suc dom 𝐺)
2624, 25mpbi 229 . . . . . . . . . . . 12 Ord suc dom 𝐺
27 ordelon 6388 . . . . . . . . . . . 12 ((Ord suc dom 𝐺 ∧ 𝐾 ∈ suc dom 𝐺) β†’ 𝐾 ∈ On)
2826, 23, 27sylancr 586 . . . . . . . . . . 11 (πœ‘ β†’ 𝐾 ∈ On)
29 ordsssuc 6453 . . . . . . . . . . 11 ((𝐾 ∈ On ∧ Ord dom 𝐺) β†’ (𝐾 βŠ† dom 𝐺 ↔ 𝐾 ∈ suc dom 𝐺))
3028, 24, 29sylancl 585 . . . . . . . . . 10 (πœ‘ β†’ (𝐾 βŠ† dom 𝐺 ↔ 𝐾 ∈ suc dom 𝐺))
3123, 30mpbird 257 . . . . . . . . 9 (πœ‘ β†’ 𝐾 βŠ† dom 𝐺)
3231adantr 480 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ 𝐾 βŠ† dom 𝐺)
33 vex 3477 . . . . . . . . . 10 π‘₯ ∈ V
3433sucid 6446 . . . . . . . . 9 π‘₯ ∈ suc π‘₯
35 simprr 770 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ 𝐾 = suc π‘₯)
3634, 35eleqtrrid 2839 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ π‘₯ ∈ 𝐾)
37 fnfvima 7237 . . . . . . . 8 ((𝐺 Fn dom 𝐺 ∧ 𝐾 βŠ† dom 𝐺 ∧ π‘₯ ∈ 𝐾) β†’ (πΊβ€˜π‘₯) ∈ (𝐺 β€œ 𝐾))
3822, 32, 36, 37syl3anc 1370 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (πΊβ€˜π‘₯) ∈ (𝐺 β€œ 𝐾))
3918, 38sseldd 3983 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (πΊβ€˜π‘₯) ∈ 𝐢)
40 ordsucss 7810 . . . . . 6 (Ord 𝐢 β†’ ((πΊβ€˜π‘₯) ∈ 𝐢 β†’ suc (πΊβ€˜π‘₯) βŠ† 𝐢))
4116, 39, 40sylc 65 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ suc (πΊβ€˜π‘₯) βŠ† 𝐢)
42 suppssdm 8166 . . . . . . . . . . 11 (𝐹 supp βˆ…) βŠ† dom 𝐹
43 cantnfcl.f . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹 ∈ 𝑆)
44 cantnfs.s . . . . . . . . . . . . . 14 𝑆 = dom (𝐴 CNF 𝐡)
45 cantnfs.b . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐡 ∈ On)
4644, 1, 45cantnfs 9665 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝐹 ∈ 𝑆 ↔ (𝐹:𝐡⟢𝐴 ∧ 𝐹 finSupp βˆ…)))
4743, 46mpbid 231 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐹:𝐡⟢𝐴 ∧ 𝐹 finSupp βˆ…))
4847simpld 494 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:𝐡⟢𝐴)
4942, 48fssdm 6737 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 supp βˆ…) βŠ† 𝐡)
50 onss 7776 . . . . . . . . . . 11 (𝐡 ∈ On β†’ 𝐡 βŠ† On)
5145, 50syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐡 βŠ† On)
5249, 51sstrd 3992 . . . . . . . . 9 (πœ‘ β†’ (𝐹 supp βˆ…) βŠ† On)
5352adantr 480 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (𝐹 supp βˆ…) βŠ† On)
5423adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ 𝐾 ∈ suc dom 𝐺)
5535, 54eqeltrrd 2833 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ suc π‘₯ ∈ suc dom 𝐺)
56 ordsucelsuc 7814 . . . . . . . . . . 11 (Ord dom 𝐺 β†’ (π‘₯ ∈ dom 𝐺 ↔ suc π‘₯ ∈ suc dom 𝐺))
5724, 56ax-mp 5 . . . . . . . . . 10 (π‘₯ ∈ dom 𝐺 ↔ suc π‘₯ ∈ suc dom 𝐺)
5855, 57sylibr 233 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ π‘₯ ∈ dom 𝐺)
5920ffvelcdmi 7085 . . . . . . . . 9 (π‘₯ ∈ dom 𝐺 β†’ (πΊβ€˜π‘₯) ∈ (𝐹 supp βˆ…))
6058, 59syl 17 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (πΊβ€˜π‘₯) ∈ (𝐹 supp βˆ…))
6153, 60sseldd 3983 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (πΊβ€˜π‘₯) ∈ On)
62 onsuc 7803 . . . . . . 7 ((πΊβ€˜π‘₯) ∈ On β†’ suc (πΊβ€˜π‘₯) ∈ On)
6361, 62syl 17 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ suc (πΊβ€˜π‘₯) ∈ On)
641adantr 480 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ 𝐴 ∈ On)
653adantr 480 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ βˆ… ∈ 𝐴)
66 oewordi 8595 . . . . . 6 (((suc (πΊβ€˜π‘₯) ∈ On ∧ 𝐢 ∈ On ∧ 𝐴 ∈ On) ∧ βˆ… ∈ 𝐴) β†’ (suc (πΊβ€˜π‘₯) βŠ† 𝐢 β†’ (𝐴 ↑o suc (πΊβ€˜π‘₯)) βŠ† (𝐴 ↑o 𝐢)))
6763, 14, 64, 65, 66syl31anc 1372 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (suc (πΊβ€˜π‘₯) βŠ† 𝐢 β†’ (𝐴 ↑o suc (πΊβ€˜π‘₯)) βŠ† (𝐴 ↑o 𝐢)))
6841, 67mpd 15 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (𝐴 ↑o suc (πΊβ€˜π‘₯)) βŠ† (𝐴 ↑o 𝐢))
6935fveq2d 6895 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (π»β€˜πΎ) = (π»β€˜suc π‘₯))
70 simprl 768 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ π‘₯ ∈ Ο‰)
71 simpl 482 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ πœ‘)
72 eleq1 2820 . . . . . . . 8 (π‘₯ = βˆ… β†’ (π‘₯ ∈ dom 𝐺 ↔ βˆ… ∈ dom 𝐺))
73 suceq 6430 . . . . . . . . . 10 (π‘₯ = βˆ… β†’ suc π‘₯ = suc βˆ…)
7473fveq2d 6895 . . . . . . . . 9 (π‘₯ = βˆ… β†’ (π»β€˜suc π‘₯) = (π»β€˜suc βˆ…))
75 fveq2 6891 . . . . . . . . . . 11 (π‘₯ = βˆ… β†’ (πΊβ€˜π‘₯) = (πΊβ€˜βˆ…))
76 suceq 6430 . . . . . . . . . . 11 ((πΊβ€˜π‘₯) = (πΊβ€˜βˆ…) β†’ suc (πΊβ€˜π‘₯) = suc (πΊβ€˜βˆ…))
7775, 76syl 17 . . . . . . . . . 10 (π‘₯ = βˆ… β†’ suc (πΊβ€˜π‘₯) = suc (πΊβ€˜βˆ…))
7877oveq2d 7428 . . . . . . . . 9 (π‘₯ = βˆ… β†’ (𝐴 ↑o suc (πΊβ€˜π‘₯)) = (𝐴 ↑o suc (πΊβ€˜βˆ…)))
7974, 78eleq12d 2826 . . . . . . . 8 (π‘₯ = βˆ… β†’ ((π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯)) ↔ (π»β€˜suc βˆ…) ∈ (𝐴 ↑o suc (πΊβ€˜βˆ…))))
8072, 79imbi12d 344 . . . . . . 7 (π‘₯ = βˆ… β†’ ((π‘₯ ∈ dom 𝐺 β†’ (π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯))) ↔ (βˆ… ∈ dom 𝐺 β†’ (π»β€˜suc βˆ…) ∈ (𝐴 ↑o suc (πΊβ€˜βˆ…)))))
81 eleq1 2820 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ dom 𝐺 ↔ 𝑦 ∈ dom 𝐺))
82 suceq 6430 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ suc π‘₯ = suc 𝑦)
8382fveq2d 6895 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (π»β€˜suc π‘₯) = (π»β€˜suc 𝑦))
84 fveq2 6891 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))
85 suceq 6430 . . . . . . . . . . 11 ((πΊβ€˜π‘₯) = (πΊβ€˜π‘¦) β†’ suc (πΊβ€˜π‘₯) = suc (πΊβ€˜π‘¦))
8684, 85syl 17 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ suc (πΊβ€˜π‘₯) = suc (πΊβ€˜π‘¦))
8786oveq2d 7428 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (𝐴 ↑o suc (πΊβ€˜π‘₯)) = (𝐴 ↑o suc (πΊβ€˜π‘¦)))
8883, 87eleq12d 2826 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯)) ↔ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦))))
8981, 88imbi12d 344 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ dom 𝐺 β†’ (π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯))) ↔ (𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))))
90 eleq1 2820 . . . . . . . 8 (π‘₯ = suc 𝑦 β†’ (π‘₯ ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
91 suceq 6430 . . . . . . . . . 10 (π‘₯ = suc 𝑦 β†’ suc π‘₯ = suc suc 𝑦)
9291fveq2d 6895 . . . . . . . . 9 (π‘₯ = suc 𝑦 β†’ (π»β€˜suc π‘₯) = (π»β€˜suc suc 𝑦))
93 fveq2 6891 . . . . . . . . . . 11 (π‘₯ = suc 𝑦 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜suc 𝑦))
94 suceq 6430 . . . . . . . . . . 11 ((πΊβ€˜π‘₯) = (πΊβ€˜suc 𝑦) β†’ suc (πΊβ€˜π‘₯) = suc (πΊβ€˜suc 𝑦))
9593, 94syl 17 . . . . . . . . . 10 (π‘₯ = suc 𝑦 β†’ suc (πΊβ€˜π‘₯) = suc (πΊβ€˜suc 𝑦))
9695oveq2d 7428 . . . . . . . . 9 (π‘₯ = suc 𝑦 β†’ (𝐴 ↑o suc (πΊβ€˜π‘₯)) = (𝐴 ↑o suc (πΊβ€˜suc 𝑦)))
9792, 96eleq12d 2826 . . . . . . . 8 (π‘₯ = suc 𝑦 β†’ ((π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯)) ↔ (π»β€˜suc suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜suc 𝑦))))
9890, 97imbi12d 344 . . . . . . 7 (π‘₯ = suc 𝑦 β†’ ((π‘₯ ∈ dom 𝐺 β†’ (π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯))) ↔ (suc 𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜suc 𝑦)))))
9948adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ 𝐹:𝐡⟢𝐴)
10020ffvelcdmi 7085 . . . . . . . . . . . 12 (βˆ… ∈ dom 𝐺 β†’ (πΊβ€˜βˆ…) ∈ (𝐹 supp βˆ…))
10149sselda 3982 . . . . . . . . . . . 12 ((πœ‘ ∧ (πΊβ€˜βˆ…) ∈ (𝐹 supp βˆ…)) β†’ (πΊβ€˜βˆ…) ∈ 𝐡)
102100, 101sylan2 592 . . . . . . . . . . 11 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (πΊβ€˜βˆ…) ∈ 𝐡)
10399, 102ffvelcdmd 7087 . . . . . . . . . 10 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (πΉβ€˜(πΊβ€˜βˆ…)) ∈ 𝐴)
1041adantr 480 . . . . . . . . . . . 12 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ 𝐴 ∈ On)
105 onelon 6389 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (πΉβ€˜(πΊβ€˜βˆ…)) ∈ 𝐴) β†’ (πΉβ€˜(πΊβ€˜βˆ…)) ∈ On)
106104, 103, 105syl2anc 583 . . . . . . . . . . 11 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (πΉβ€˜(πΊβ€˜βˆ…)) ∈ On)
10752sselda 3982 . . . . . . . . . . . . 13 ((πœ‘ ∧ (πΊβ€˜βˆ…) ∈ (𝐹 supp βˆ…)) β†’ (πΊβ€˜βˆ…) ∈ On)
108100, 107sylan2 592 . . . . . . . . . . . 12 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (πΊβ€˜βˆ…) ∈ On)
109 oecl 8541 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (πΊβ€˜βˆ…) ∈ On) β†’ (𝐴 ↑o (πΊβ€˜βˆ…)) ∈ On)
110104, 108, 109syl2anc 583 . . . . . . . . . . 11 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (𝐴 ↑o (πΊβ€˜βˆ…)) ∈ On)
1113adantr 480 . . . . . . . . . . . 12 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ βˆ… ∈ 𝐴)
112 oen0 8590 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (πΊβ€˜βˆ…) ∈ On) ∧ βˆ… ∈ 𝐴) β†’ βˆ… ∈ (𝐴 ↑o (πΊβ€˜βˆ…)))
113104, 108, 111, 112syl21anc 835 . . . . . . . . . . 11 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ βˆ… ∈ (𝐴 ↑o (πΊβ€˜βˆ…)))
114 omord2 8571 . . . . . . . . . . 11 ((((πΉβ€˜(πΊβ€˜βˆ…)) ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ↑o (πΊβ€˜βˆ…)) ∈ On) ∧ βˆ… ∈ (𝐴 ↑o (πΊβ€˜βˆ…))) β†’ ((πΉβ€˜(πΊβ€˜βˆ…)) ∈ 𝐴 ↔ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) ∈ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o 𝐴)))
115106, 104, 110, 113, 114syl31anc 1372 . . . . . . . . . 10 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ ((πΉβ€˜(πΊβ€˜βˆ…)) ∈ 𝐴 ↔ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) ∈ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o 𝐴)))
116103, 115mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) ∈ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o 𝐴))
117 peano1 7883 . . . . . . . . . . . 12 βˆ… ∈ Ο‰
118117a1i 11 . . . . . . . . . . 11 (βˆ… ∈ dom 𝐺 β†’ βˆ… ∈ Ο‰)
11944, 1, 45, 19, 43, 8cantnfsuc 9669 . . . . . . . . . . 11 ((πœ‘ ∧ βˆ… ∈ Ο‰) β†’ (π»β€˜suc βˆ…) = (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) +o (π»β€˜βˆ…)))
120118, 119sylan2 592 . . . . . . . . . 10 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (π»β€˜suc βˆ…) = (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) +o (π»β€˜βˆ…)))
12110oveq2i 7423 . . . . . . . . . . 11 (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) +o (π»β€˜βˆ…)) = (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) +o βˆ…)
122 omcl 8540 . . . . . . . . . . . . 13 (((𝐴 ↑o (πΊβ€˜βˆ…)) ∈ On ∧ (πΉβ€˜(πΊβ€˜βˆ…)) ∈ On) β†’ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) ∈ On)
123110, 106, 122syl2anc 583 . . . . . . . . . . . 12 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) ∈ On)
124 oa0 8520 . . . . . . . . . . . 12 (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) ∈ On β†’ (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) +o βˆ…) = ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))))
125123, 124syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) +o βˆ…) = ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))))
126121, 125eqtrid 2783 . . . . . . . . . 10 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))) +o (π»β€˜βˆ…)) = ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))))
127120, 126eqtrd 2771 . . . . . . . . 9 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (π»β€˜suc βˆ…) = ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o (πΉβ€˜(πΊβ€˜βˆ…))))
128 oesuc 8531 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (πΊβ€˜βˆ…) ∈ On) β†’ (𝐴 ↑o suc (πΊβ€˜βˆ…)) = ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o 𝐴))
129104, 108, 128syl2anc 583 . . . . . . . . 9 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (𝐴 ↑o suc (πΊβ€˜βˆ…)) = ((𝐴 ↑o (πΊβ€˜βˆ…)) Β·o 𝐴))
130116, 127, 1293eltr4d 2847 . . . . . . . 8 ((πœ‘ ∧ βˆ… ∈ dom 𝐺) β†’ (π»β€˜suc βˆ…) ∈ (𝐴 ↑o suc (πΊβ€˜βˆ…)))
131130ex 412 . . . . . . 7 (πœ‘ β†’ (βˆ… ∈ dom 𝐺 β†’ (π»β€˜suc βˆ…) ∈ (𝐴 ↑o suc (πΊβ€˜βˆ…))))
132 ordtr 6378 . . . . . . . . . . . 12 (Ord dom 𝐺 β†’ Tr dom 𝐺)
13324, 132ax-mp 5 . . . . . . . . . . 11 Tr dom 𝐺
134 trsuc 6451 . . . . . . . . . . 11 ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) β†’ 𝑦 ∈ dom 𝐺)
135133, 134mpan 687 . . . . . . . . . 10 (suc 𝑦 ∈ dom 𝐺 β†’ 𝑦 ∈ dom 𝐺)
136135imim1i 63 . . . . . . . . 9 ((𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦))) β†’ (suc 𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦))))
1371ad2antrr 723 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ 𝐴 ∈ On)
138 eloni 6374 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On β†’ Ord 𝐴)
139137, 138syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ Ord 𝐴)
14048ad2antrr 723 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ 𝐹:𝐡⟢𝐴)
14149ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (𝐹 supp βˆ…) βŠ† 𝐡)
14220ffvelcdmi 7085 . . . . . . . . . . . . . . . . . 18 (suc 𝑦 ∈ dom 𝐺 β†’ (πΊβ€˜suc 𝑦) ∈ (𝐹 supp βˆ…))
143142ad2antrl 725 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΊβ€˜suc 𝑦) ∈ (𝐹 supp βˆ…))
144141, 143sseldd 3983 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΊβ€˜suc 𝑦) ∈ 𝐡)
145140, 144ffvelcdmd 7087 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ 𝐴)
146 ordsucss 7810 . . . . . . . . . . . . . . 15 (Ord 𝐴 β†’ ((πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ 𝐴 β†’ suc (πΉβ€˜(πΊβ€˜suc 𝑦)) βŠ† 𝐴))
147139, 145, 146sylc 65 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ suc (πΉβ€˜(πΊβ€˜suc 𝑦)) βŠ† 𝐴)
148 onelon 6389 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ 𝐴) β†’ (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On)
149137, 145, 148syl2anc 583 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On)
150 onsuc 7803 . . . . . . . . . . . . . . . 16 ((πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On β†’ suc (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On)
151149, 150syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ suc (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On)
15252ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (𝐹 supp βˆ…) βŠ† On)
153152, 143sseldd 3983 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΊβ€˜suc 𝑦) ∈ On)
154 oecl 8541 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ (πΊβ€˜suc 𝑦) ∈ On) β†’ (𝐴 ↑o (πΊβ€˜suc 𝑦)) ∈ On)
155137, 153, 154syl2anc 583 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (𝐴 ↑o (πΊβ€˜suc 𝑦)) ∈ On)
156 omwordi 8575 . . . . . . . . . . . . . . 15 ((suc (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ↑o (πΊβ€˜suc 𝑦)) ∈ On) β†’ (suc (πΉβ€˜(πΊβ€˜suc 𝑦)) βŠ† 𝐴 β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o suc (πΉβ€˜(πΊβ€˜suc 𝑦))) βŠ† ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o 𝐴)))
157151, 137, 155, 156syl3anc 1370 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (suc (πΉβ€˜(πΊβ€˜suc 𝑦)) βŠ† 𝐴 β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o suc (πΉβ€˜(πΊβ€˜suc 𝑦))) βŠ† ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o 𝐴)))
158147, 157mpd 15 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o suc (πΉβ€˜(πΊβ€˜suc 𝑦))) βŠ† ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o 𝐴))
159 oesuc 8531 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (πΊβ€˜suc 𝑦) ∈ On) β†’ (𝐴 ↑o suc (πΊβ€˜suc 𝑦)) = ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o 𝐴))
160137, 153, 159syl2anc 583 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (𝐴 ↑o suc (πΊβ€˜suc 𝑦)) = ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o 𝐴))
161158, 160sseqtrrd 4023 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o suc (πΉβ€˜(πΊβ€˜suc 𝑦))) βŠ† (𝐴 ↑o suc (πΊβ€˜suc 𝑦)))
162 eloni 6374 . . . . . . . . . . . . . . . . . 18 ((πΊβ€˜suc 𝑦) ∈ On β†’ Ord (πΊβ€˜suc 𝑦))
163153, 162syl 17 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ Ord (πΊβ€˜suc 𝑦))
164 vex 3477 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
165164sucid 6446 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ suc 𝑦
166164sucex 7798 . . . . . . . . . . . . . . . . . . . . 21 suc 𝑦 ∈ V
167166epeli 5582 . . . . . . . . . . . . . . . . . . . 20 (𝑦 E suc 𝑦 ↔ 𝑦 ∈ suc 𝑦)
168165, 167mpbir 230 . . . . . . . . . . . . . . . . . . 19 𝑦 E suc 𝑦
169 ovexd 7447 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (𝐹 supp βˆ…) ∈ V)
17044, 1, 45, 19, 43cantnfcl 9666 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ ( E We (𝐹 supp βˆ…) ∧ dom 𝐺 ∈ Ο‰))
171170simpld 494 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ E We (𝐹 supp βˆ…))
17219oiiso 9536 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 supp βˆ…) ∈ V ∧ E We (𝐹 supp βˆ…)) β†’ 𝐺 Isom E , E (dom 𝐺, (𝐹 supp βˆ…)))
173169, 171, 172syl2anc 583 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ 𝐺 Isom E , E (dom 𝐺, (𝐹 supp βˆ…)))
174173ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ 𝐺 Isom E , E (dom 𝐺, (𝐹 supp βˆ…)))
175135ad2antrl 725 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ 𝑦 ∈ dom 𝐺)
176 simprl 768 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ suc 𝑦 ∈ dom 𝐺)
177 isorel 7326 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp βˆ…)) ∧ (𝑦 ∈ dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺)) β†’ (𝑦 E suc 𝑦 ↔ (πΊβ€˜π‘¦) E (πΊβ€˜suc 𝑦)))
178174, 175, 176, 177syl12anc 834 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (𝑦 E suc 𝑦 ↔ (πΊβ€˜π‘¦) E (πΊβ€˜suc 𝑦)))
179168, 178mpbii 232 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΊβ€˜π‘¦) E (πΊβ€˜suc 𝑦))
180 fvex 6904 . . . . . . . . . . . . . . . . . . 19 (πΊβ€˜suc 𝑦) ∈ V
181180epeli 5582 . . . . . . . . . . . . . . . . . 18 ((πΊβ€˜π‘¦) E (πΊβ€˜suc 𝑦) ↔ (πΊβ€˜π‘¦) ∈ (πΊβ€˜suc 𝑦))
182179, 181sylib 217 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΊβ€˜π‘¦) ∈ (πΊβ€˜suc 𝑦))
183 ordsucss 7810 . . . . . . . . . . . . . . . . 17 (Ord (πΊβ€˜suc 𝑦) β†’ ((πΊβ€˜π‘¦) ∈ (πΊβ€˜suc 𝑦) β†’ suc (πΊβ€˜π‘¦) βŠ† (πΊβ€˜suc 𝑦)))
184163, 182, 183sylc 65 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ suc (πΊβ€˜π‘¦) βŠ† (πΊβ€˜suc 𝑦))
18520ffvelcdmi 7085 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ dom 𝐺 β†’ (πΊβ€˜π‘¦) ∈ (𝐹 supp βˆ…))
186175, 185syl 17 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΊβ€˜π‘¦) ∈ (𝐹 supp βˆ…))
187152, 186sseldd 3983 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (πΊβ€˜π‘¦) ∈ On)
188 onsuc 7803 . . . . . . . . . . . . . . . . . 18 ((πΊβ€˜π‘¦) ∈ On β†’ suc (πΊβ€˜π‘¦) ∈ On)
189187, 188syl 17 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ suc (πΊβ€˜π‘¦) ∈ On)
1903ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ βˆ… ∈ 𝐴)
191 oewordi 8595 . . . . . . . . . . . . . . . . 17 (((suc (πΊβ€˜π‘¦) ∈ On ∧ (πΊβ€˜suc 𝑦) ∈ On ∧ 𝐴 ∈ On) ∧ βˆ… ∈ 𝐴) β†’ (suc (πΊβ€˜π‘¦) βŠ† (πΊβ€˜suc 𝑦) β†’ (𝐴 ↑o suc (πΊβ€˜π‘¦)) βŠ† (𝐴 ↑o (πΊβ€˜suc 𝑦))))
192189, 153, 137, 190, 191syl31anc 1372 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (suc (πΊβ€˜π‘¦) βŠ† (πΊβ€˜suc 𝑦) β†’ (𝐴 ↑o suc (πΊβ€˜π‘¦)) βŠ† (𝐴 ↑o (πΊβ€˜suc 𝑦))))
193184, 192mpd 15 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (𝐴 ↑o suc (πΊβ€˜π‘¦)) βŠ† (𝐴 ↑o (πΊβ€˜suc 𝑦)))
194 simprr 770 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))
195193, 194sseldd 3983 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o (πΊβ€˜suc 𝑦)))
196 peano2 7885 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ Ο‰ β†’ suc 𝑦 ∈ Ο‰)
197196ad2antlr 724 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ suc 𝑦 ∈ Ο‰)
1988cantnfvalf 9664 . . . . . . . . . . . . . . . . 17 𝐻:Ο‰βŸΆOn
199198ffvelcdmi 7085 . . . . . . . . . . . . . . . 16 (suc 𝑦 ∈ Ο‰ β†’ (π»β€˜suc 𝑦) ∈ On)
200197, 199syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (π»β€˜suc 𝑦) ∈ On)
201 omcl 8540 . . . . . . . . . . . . . . . 16 (((𝐴 ↑o (πΊβ€˜suc 𝑦)) ∈ On ∧ (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On) β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) ∈ On)
202155, 149, 201syl2anc 583 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) ∈ On)
203 oaord 8551 . . . . . . . . . . . . . . 15 (((π»β€˜suc 𝑦) ∈ On ∧ (𝐴 ↑o (πΊβ€˜suc 𝑦)) ∈ On ∧ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) ∈ On) β†’ ((π»β€˜suc 𝑦) ∈ (𝐴 ↑o (πΊβ€˜suc 𝑦)) ↔ (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (π»β€˜suc 𝑦)) ∈ (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (𝐴 ↑o (πΊβ€˜suc 𝑦)))))
204200, 155, 202, 203syl3anc 1370 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ ((π»β€˜suc 𝑦) ∈ (𝐴 ↑o (πΊβ€˜suc 𝑦)) ↔ (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (π»β€˜suc 𝑦)) ∈ (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (𝐴 ↑o (πΊβ€˜suc 𝑦)))))
205195, 204mpbid 231 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (π»β€˜suc 𝑦)) ∈ (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (𝐴 ↑o (πΊβ€˜suc 𝑦))))
20644, 1, 45, 19, 43, 8cantnfsuc 9669 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ suc 𝑦 ∈ Ο‰) β†’ (π»β€˜suc suc 𝑦) = (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (π»β€˜suc 𝑦)))
207196, 206sylan2 592 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ Ο‰) β†’ (π»β€˜suc suc 𝑦) = (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (π»β€˜suc 𝑦)))
208207adantr 480 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (π»β€˜suc suc 𝑦) = (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (π»β€˜suc 𝑦)))
209 omsuc 8530 . . . . . . . . . . . . . 14 (((𝐴 ↑o (πΊβ€˜suc 𝑦)) ∈ On ∧ (πΉβ€˜(πΊβ€˜suc 𝑦)) ∈ On) β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o suc (πΉβ€˜(πΊβ€˜suc 𝑦))) = (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (𝐴 ↑o (πΊβ€˜suc 𝑦))))
210155, 149, 209syl2anc 583 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o suc (πΉβ€˜(πΊβ€˜suc 𝑦))) = (((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o (πΉβ€˜(πΊβ€˜suc 𝑦))) +o (𝐴 ↑o (πΊβ€˜suc 𝑦))))
211205, 208, 2103eltr4d 2847 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (π»β€˜suc suc 𝑦) ∈ ((𝐴 ↑o (πΊβ€˜suc 𝑦)) Β·o suc (πΉβ€˜(πΊβ€˜suc 𝑦))))
212161, 211sseldd 3983 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ Ο‰) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)))) β†’ (π»β€˜suc suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜suc 𝑦)))
213212exp32 420 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ Ο‰) β†’ (suc 𝑦 ∈ dom 𝐺 β†’ ((π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦)) β†’ (π»β€˜suc suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜suc 𝑦)))))
214213a2d 29 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ Ο‰) β†’ ((suc 𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦))) β†’ (suc 𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜suc 𝑦)))))
215136, 214syl5 34 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ Ο‰) β†’ ((𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦))) β†’ (suc 𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜suc 𝑦)))))
216215expcom 413 . . . . . . 7 (𝑦 ∈ Ο‰ β†’ (πœ‘ β†’ ((𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜π‘¦))) β†’ (suc 𝑦 ∈ dom 𝐺 β†’ (π»β€˜suc suc 𝑦) ∈ (𝐴 ↑o suc (πΊβ€˜suc 𝑦))))))
21780, 89, 98, 131, 216finds2 7895 . . . . . 6 (π‘₯ ∈ Ο‰ β†’ (πœ‘ β†’ (π‘₯ ∈ dom 𝐺 β†’ (π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯)))))
21870, 71, 58, 217syl3c 66 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (π»β€˜suc π‘₯) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯)))
21969, 218eqeltrd 2832 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (π»β€˜πΎ) ∈ (𝐴 ↑o suc (πΊβ€˜π‘₯)))
22068, 219sseldd 3983 . . 3 ((πœ‘ ∧ (π‘₯ ∈ Ο‰ ∧ 𝐾 = suc π‘₯)) β†’ (π»β€˜πΎ) ∈ (𝐴 ↑o 𝐢))
221220rexlimdvaa 3155 . 2 (πœ‘ β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐾 = suc π‘₯ β†’ (π»β€˜πΎ) ∈ (𝐴 ↑o 𝐢)))
222 peano2 7885 . . . . 5 (dom 𝐺 ∈ Ο‰ β†’ suc dom 𝐺 ∈ Ο‰)
223170, 222simpl2im 503 . . . 4 (πœ‘ β†’ suc dom 𝐺 ∈ Ο‰)
224 elnn 7870 . . . 4 ((𝐾 ∈ suc dom 𝐺 ∧ suc dom 𝐺 ∈ Ο‰) β†’ 𝐾 ∈ Ο‰)
22523, 223, 224syl2anc 583 . . 3 (πœ‘ β†’ 𝐾 ∈ Ο‰)
226 nn0suc 7890 . . 3 (𝐾 ∈ Ο‰ β†’ (𝐾 = βˆ… ∨ βˆƒπ‘₯ ∈ Ο‰ 𝐾 = suc π‘₯))
227225, 226syl 17 . 2 (πœ‘ β†’ (𝐾 = βˆ… ∨ βˆƒπ‘₯ ∈ Ο‰ 𝐾 = suc π‘₯))
22813, 221, 227mpjaod 857 1 (πœ‘ β†’ (π»β€˜πΎ) ∈ (𝐴 ↑o 𝐢))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105  βˆƒwrex 3069  Vcvv 3473   βŠ† wss 3948  βˆ…c0 4322   class class class wbr 5148  Tr wtr 5265   E cep 5579   We wwe 5630  dom cdm 5676   β€œ cima 5679  Ord word 6363  Oncon0 6364  suc csuc 6366   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543   Isom wiso 6544  (class class class)co 7412   ∈ cmpo 7414  Ο‰com 7859   supp csupp 8150  seqΟ‰cseqom 8451   +o coa 8467   Β·o comu 8468   ↑o coe 8469   finSupp cfsupp 9365  OrdIsocoi 9508   CNF ccnf 9660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8151  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-seqom 8452  df-1o 8470  df-2o 8471  df-oadd 8474  df-omul 8475  df-oexp 8476  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9366  df-oi 9509  df-cnf 9661
This theorem is referenced by:  cantnflt2  9672  cnfcomlem  9698
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