MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnfcomlem Structured version   Visualization version   GIF version

Theorem cnfcomlem 8852
Description: Lemma for cnfcom 8853. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
Hypotheses
Ref Expression
cnfcom.s 𝑆 = dom (ω CNF 𝐴)
cnfcom.a (𝜑𝐴 ∈ On)
cnfcom.b (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
cnfcom.f 𝐹 = ((ω CNF 𝐴)‘𝐵)
cnfcom.g 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
cnfcom.t 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
cnfcom.k 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
cnfcom.1 (𝜑𝐼 ∈ dom 𝐺)
cnfcom.2 (𝜑𝑂 ∈ (ω ↑𝑜 (𝐺𝐼)))
cnfcom.3 (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)
Assertion
Ref Expression
cnfcomlem (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
Distinct variable groups:   𝑥,𝑘,𝑧,𝐴   𝑘,𝐼,𝑥,𝑧   𝑥,𝑀   𝑓,𝑘,𝑥,𝑧,𝐹   𝑧,𝑇   𝑓,𝐺,𝑘,𝑥,𝑧   𝑓,𝐻,𝑥   𝑆,𝑘,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑓,𝑘)   𝐴(𝑓)   𝐵(𝑥,𝑧,𝑓,𝑘)   𝑆(𝑥,𝑓)   𝑇(𝑥,𝑓,𝑘)   𝐻(𝑧,𝑘)   𝐼(𝑓)   𝐾(𝑥,𝑧,𝑓,𝑘)   𝑀(𝑧,𝑓,𝑘)   𝑂(𝑥,𝑧,𝑓,𝑘)

Proof of Theorem cnfcomlem
Dummy variables 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omelon 8799 . . . . . . 7 ω ∈ On
2 cnfcom.a . . . . . . . 8 (𝜑𝐴 ∈ On)
3 suppssdm 7551 . . . . . . . . . 10 (𝐹 supp ∅) ⊆ dom 𝐹
4 cnfcom.f . . . . . . . . . . . . 13 𝐹 = ((ω CNF 𝐴)‘𝐵)
5 cnfcom.s . . . . . . . . . . . . . . . 16 𝑆 = dom (ω CNF 𝐴)
61a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ω ∈ On)
75, 6, 2cantnff1o 8849 . . . . . . . . . . . . . . 15 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
8 f1ocnv 6374 . . . . . . . . . . . . . . 15 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
9 f1of 6362 . . . . . . . . . . . . . . 15 ((ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
107, 8, 93syl 18 . . . . . . . . . . . . . 14 (𝜑(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
11 cnfcom.b . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
1210, 11ffvelrnd 6591 . . . . . . . . . . . . 13 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
134, 12syl5eqel 2900 . . . . . . . . . . . 12 (𝜑𝐹𝑆)
145, 6, 2cantnfs 8819 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
1513, 14mpbid 223 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
1615simpld 484 . . . . . . . . . 10 (𝜑𝐹:𝐴⟶ω)
173, 16fssdm 6281 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
18 cnfcom.1 . . . . . . . . . 10 (𝜑𝐼 ∈ dom 𝐺)
19 cnfcom.g . . . . . . . . . . . 12 𝐺 = OrdIso( E , (𝐹 supp ∅))
2019oif 8683 . . . . . . . . . . 11 𝐺:dom 𝐺⟶(𝐹 supp ∅)
2120ffvelrni 6589 . . . . . . . . . 10 (𝐼 ∈ dom 𝐺 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2218, 21syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝐼) ∈ (𝐹 supp ∅))
2317, 22sseldd 3810 . . . . . . . 8 (𝜑 → (𝐺𝐼) ∈ 𝐴)
24 onelon 5974 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼) ∈ On)
252, 23, 24syl2anc 575 . . . . . . 7 (𝜑 → (𝐺𝐼) ∈ On)
26 oecl 7863 . . . . . . 7 ((ω ∈ On ∧ (𝐺𝐼) ∈ On) → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
271, 25, 26sylancr 577 . . . . . 6 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ∈ On)
2816, 23ffvelrnd 6591 . . . . . . 7 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ ω)
29 nnon 7310 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ ω → (𝐹‘(𝐺𝐼)) ∈ On)
3028, 29syl 17 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ∈ On)
31 omcl 7862 . . . . . 6 (((ω ↑𝑜 (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) → ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On)
3227, 30, 31syl2anc 575 . . . . 5 (𝜑 → ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On)
335, 6, 2, 19, 13cantnfcl 8820 . . . . . . . 8 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
3433simprd 485 . . . . . . 7 (𝜑 → dom 𝐺 ∈ ω)
35 elnn 7314 . . . . . . 7 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
3618, 34, 35syl2anc 575 . . . . . 6 (𝜑𝐼 ∈ ω)
37 cnfcom.h . . . . . . . 8 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
3837cantnfvalf 8818 . . . . . . 7 𝐻:ω⟶On
3938ffvelrni 6589 . . . . . 6 (𝐼 ∈ ω → (𝐻𝐼) ∈ On)
4036, 39syl 17 . . . . 5 (𝜑 → (𝐻𝐼) ∈ On)
41 eqid 2817 . . . . . 6 ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)))
4241oacomf1o 7891 . . . . 5 ((((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On ∧ (𝐻𝐼) ∈ On) → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
4332, 40, 42syl2anc 575 . . . 4 (𝜑 → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
44 cnfcom.t . . . . . . . 8 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
4544seqomsuc 7797 . . . . . . 7 (𝐼 ∈ ω → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)))
4636, 45syl 17 . . . . . 6 (𝜑 → (𝑇‘suc 𝐼) = (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)))
47 nfcv 2959 . . . . . . . . 9 𝑢𝐾
48 nfcv 2959 . . . . . . . . 9 𝑣𝐾
49 nfcv 2959 . . . . . . . . 9 𝑘((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
50 nfcv 2959 . . . . . . . . 9 𝑓((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
51 cnfcom.k . . . . . . . . . 10 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
52 oveq2 6891 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (dom 𝑓 +𝑜 𝑥) = (dom 𝑓 +𝑜 𝑦))
5352cbvmptv 4955 . . . . . . . . . . . 12 (𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦𝑀 ↦ (dom 𝑓 +𝑜 𝑦))
54 cnfcom.m . . . . . . . . . . . . . 14 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
55 simpl 470 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑘 = 𝑢)
5655fveq2d 6421 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐺𝑘) = (𝐺𝑢))
5756oveq2d 6899 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (ω ↑𝑜 (𝐺𝑘)) = (ω ↑𝑜 (𝐺𝑢)))
5856fveq2d 6421 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
5957, 58oveq12d 6901 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
6054, 59syl5eq 2863 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑀 = ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
61 simpr 473 . . . . . . . . . . . . . . 15 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝑓 = 𝑣)
6261dmeqd 5540 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → dom 𝑓 = dom 𝑣)
6362oveq1d 6898 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (dom 𝑓 +𝑜 𝑦) = (dom 𝑣 +𝑜 𝑦))
6460, 63mpteq12dv 4938 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦𝑀 ↦ (dom 𝑓 +𝑜 𝑦)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
6553, 64syl5eq 2863 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)))
66 oveq2 6891 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 𝑦))
6766cbvmptv 4955 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦))
6860oveq1d 6898 . . . . . . . . . . . . . 14 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑀 +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))
6962, 68mpteq12dv 4938 . . . . . . . . . . . . 13 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑦 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑦)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7067, 69syl5eq 2863 . . . . . . . . . . . 12 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7170cnveqd 5512 . . . . . . . . . . 11 ((𝑘 = 𝑢𝑓 = 𝑣) → (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)) = (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))
7265, 71uneq12d 3978 . . . . . . . . . 10 ((𝑘 = 𝑢𝑓 = 𝑣) → ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥))) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))))
7351, 72syl5eq 2863 . . . . . . . . 9 ((𝑘 = 𝑢𝑓 = 𝑣) → 𝐾 = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))))
7447, 48, 49, 50, 73cbvmpt2 6973 . . . . . . . 8 (𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾) = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))))
7574a1i 11 . . . . . . 7 (𝜑 → (𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾) = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)))))
76 simprl 778 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → 𝑢 = 𝐼)
7776fveq2d 6421 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐺𝑢) = (𝐺𝐼))
7877oveq2d 6899 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (ω ↑𝑜 (𝐺𝑢)) = (ω ↑𝑜 (𝐺𝐼)))
7977fveq2d 6421 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐼)))
8078, 79oveq12d 6901 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
81 simpr 473 . . . . . . . . . . . 12 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → 𝑣 = (𝑇𝐼))
8281dmeqd 5540 . . . . . . . . . . 11 ((𝑢 = 𝐼𝑣 = (𝑇𝐼)) → dom 𝑣 = dom (𝑇𝐼))
83 cnfcom.3 . . . . . . . . . . . 12 (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)
84 f1odm 6366 . . . . . . . . . . . 12 ((𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂 → dom (𝑇𝐼) = (𝐻𝐼))
8583, 84syl 17 . . . . . . . . . . 11 (𝜑 → dom (𝑇𝐼) = (𝐻𝐼))
8682, 85sylan9eqr 2873 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → dom 𝑣 = (𝐻𝐼))
8786oveq1d 6898 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (dom 𝑣 +𝑜 𝑦) = ((𝐻𝐼) +𝑜 𝑦))
8880, 87mpteq12dv 4938 . . . . . . . 8 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) = (𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)))
8980oveq1d 6898 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))
9086, 89mpteq12dv 4938 . . . . . . . . 9 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)) = (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)))
9190cnveqd 5512 . . . . . . . 8 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦)) = (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)))
9288, 91uneq12d 3978 . . . . . . 7 ((𝜑 ∧ (𝑢 = 𝐼𝑣 = (𝑇𝐼))) → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) ↦ (dom 𝑣 +𝑜 𝑦)) ∪ (𝑦 ∈ dom 𝑣 ↦ (((ω ↑𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑦))) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))))
93 elex 3417 . . . . . . . 8 (𝐼 ∈ dom 𝐺𝐼 ∈ V)
9418, 93syl 17 . . . . . . 7 (𝜑𝐼 ∈ V)
95 fvexd 6432 . . . . . . 7 (𝜑 → (𝑇𝐼) ∈ V)
96 ovex 6915 . . . . . . . . . 10 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ V
9796mptex 6720 . . . . . . . . 9 (𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∈ V
98 fvex 6430 . . . . . . . . . . 11 (𝐻𝐼) ∈ V
9998mptex 6720 . . . . . . . . . 10 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
10099cnvex 7352 . . . . . . . . 9 (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦)) ∈ V
10197, 100unex 7195 . . . . . . . 8 ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) ∈ V
102101a1i 11 . . . . . . 7 (𝜑 → ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) ∈ V)
10375, 92, 94, 95, 102ovmpt2d 7027 . . . . . 6 (𝜑 → (𝐼(𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾)(𝑇𝐼)) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))))
10446, 103eqtrd 2851 . . . . 5 (𝜑 → (𝑇‘suc 𝐼) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))))
105 f1oeq1 6352 . . . . 5 ((𝑇‘suc 𝐼) = ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))) → ((𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
106104, 105syl 17 . . . 4 (𝜑 → ((𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ ((𝑦 ∈ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ↦ ((𝐻𝐼) +𝑜 𝑦)) ∪ (𝑦 ∈ (𝐻𝐼) ↦ (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 𝑦))):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
10743, 106mpbird 248 . . 3 (𝜑 → (𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
1081a1i 11 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → ω ∈ On)
109 simpl 470 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐴 ∈ On)
110 simpr 473 . . . . . 6 ((𝐴 ∈ On ∧ 𝐹𝑆) → 𝐹𝑆)
11154oveq1i 6893 . . . . . . . . . 10 (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)
112111a1i 11 . . . . . . . . 9 ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
113112mpt2eq3ia 6959 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
114 eqid 2817 . . . . . . . 8 ∅ = ∅
115 seqomeq12 7794 . . . . . . . 8 (((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅))
116113, 114, 115mp2an 675 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
11737, 116eqtri 2839 . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
1185, 108, 109, 19, 110, 117cantnfsuc 8823 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ 𝐼 ∈ ω) → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
1192, 13, 36, 118syl21anc 857 . . . 4 (𝜑 → (𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)))
120 f1oeq2 6353 . . . 4 ((𝐻‘suc 𝐼) = (((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼)) → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
121119, 120syl 17 . . 3 (𝜑 → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) +𝑜 (𝐻𝐼))–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
122107, 121mpbird 248 . 2 (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
123 sssucid 6027 . . . . . 6 dom 𝐺 ⊆ suc dom 𝐺
124123, 18sseldi 3807 . . . . 5 (𝜑𝐼 ∈ suc dom 𝐺)
125 epelg 5238 . . . . . . . . . . 11 (𝐼 ∈ dom 𝐺 → (𝑦 E 𝐼𝑦𝐼))
12618, 125syl 17 . . . . . . . . . 10 (𝜑 → (𝑦 E 𝐼𝑦𝐼))
127126biimpar 465 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝑦 E 𝐼)
128 ovexd 6917 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp ∅) ∈ V)
12933simpld 484 . . . . . . . . . . . 12 (𝜑 → E We (𝐹 supp ∅))
13019oiiso 8690 . . . . . . . . . . . 12 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
131128, 129, 130syl2anc 575 . . . . . . . . . . 11 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
132131adantr 468 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
13319oicl 8682 . . . . . . . . . . . 12 Ord dom 𝐺
134 ordelss 5965 . . . . . . . . . . . 12 ((Ord dom 𝐺𝐼 ∈ dom 𝐺) → 𝐼 ⊆ dom 𝐺)
135133, 18, 134sylancr 577 . . . . . . . . . . 11 (𝜑𝐼 ⊆ dom 𝐺)
136135sselda 3809 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦 ∈ dom 𝐺)
13718adantr 468 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼 ∈ dom 𝐺)
138 isorel 6809 . . . . . . . . . 10 ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺𝐼 ∈ dom 𝐺)) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
139132, 136, 137, 138syl12anc 856 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑦 E 𝐼 ↔ (𝐺𝑦) E (𝐺𝐼)))
140127, 139mpbid 223 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝐺𝑦) E (𝐺𝐼))
141 fvex 6430 . . . . . . . . 9 (𝐺𝐼) ∈ V
142141epeli 5239 . . . . . . . 8 ((𝐺𝑦) E (𝐺𝐼) ↔ (𝐺𝑦) ∈ (𝐺𝐼))
143140, 142sylib 209 . . . . . . 7 ((𝜑𝑦𝐼) → (𝐺𝑦) ∈ (𝐺𝐼))
144143ralrimiva 3165 . . . . . 6 (𝜑 → ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼))
145 ffun 6268 . . . . . . . 8 (𝐺:dom 𝐺⟶(𝐹 supp ∅) → Fun 𝐺)
14620, 145ax-mp 5 . . . . . . 7 Fun 𝐺
147 funimass4 6477 . . . . . . 7 ((Fun 𝐺𝐼 ⊆ dom 𝐺) → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
148146, 135, 147sylancr 577 . . . . . 6 (𝜑 → ((𝐺𝐼) ⊆ (𝐺𝐼) ↔ ∀𝑦𝐼 (𝐺𝑦) ∈ (𝐺𝐼)))
149144, 148mpbird 248 . . . . 5 (𝜑 → (𝐺𝐼) ⊆ (𝐺𝐼))
1501a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ω ∈ On)
151 simpll 774 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐴 ∈ On)
152 simplr 776 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐹𝑆)
153 peano1 7324 . . . . . . 7 ∅ ∈ ω
154153a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → ∅ ∈ ω)
155 simpr1 1241 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → 𝐼 ∈ suc dom 𝐺)
156 simpr2 1243 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ∈ On)
157 simpr3 1245 . . . . . 6 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐺𝐼) ⊆ (𝐺𝐼))
1585, 150, 151, 19, 152, 117, 154, 155, 156, 157cantnflt 8825 . . . . 5 (((𝐴 ∈ On ∧ 𝐹𝑆) ∧ (𝐼 ∈ suc dom 𝐺 ∧ (𝐺𝐼) ∈ On ∧ (𝐺𝐼) ⊆ (𝐺𝐼))) → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
1592, 13, 124, 25, 149, 158syl23anc 1489 . . . 4 (𝜑 → (𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)))
16016ffnd 6266 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
161 0ex 4997 . . . . . . . . . 10 ∅ ∈ V
162161a1i 11 . . . . . . . . 9 (𝜑 → ∅ ∈ V)
163 elsuppfn 7546 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴 ∈ On ∧ ∅ ∈ V) → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
164160, 2, 162, 163syl3anc 1483 . . . . . . . 8 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅)))
165 simpr 473 . . . . . . . 8 (((𝐺𝐼) ∈ 𝐴 ∧ (𝐹‘(𝐺𝐼)) ≠ ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅)
166164, 165syl6bi 244 . . . . . . 7 (𝜑 → ((𝐺𝐼) ∈ (𝐹 supp ∅) → (𝐹‘(𝐺𝐼)) ≠ ∅))
16722, 166mpd 15 . . . . . 6 (𝜑 → (𝐹‘(𝐺𝐼)) ≠ ∅)
168 on0eln0 6005 . . . . . . 7 ((𝐹‘(𝐺𝐼)) ∈ On → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
16930, 168syl 17 . . . . . 6 (𝜑 → (∅ ∈ (𝐹‘(𝐺𝐼)) ↔ (𝐹‘(𝐺𝐼)) ≠ ∅))
170167, 169mpbird 248 . . . . 5 (𝜑 → ∅ ∈ (𝐹‘(𝐺𝐼)))
171 omword1 7899 . . . . 5 ((((ω ↑𝑜 (𝐺𝐼)) ∈ On ∧ (𝐹‘(𝐺𝐼)) ∈ On) ∧ ∅ ∈ (𝐹‘(𝐺𝐼))) → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
17227, 30, 170, 171syl21anc 857 . . . 4 (𝜑 → (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
173 oaabs2 7971 . . . 4 ((((𝐻𝐼) ∈ (ω ↑𝑜 (𝐺𝐼)) ∧ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) ∈ On) ∧ (ω ↑𝑜 (𝐺𝐼)) ⊆ ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) → ((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
174159, 32, 172, 173syl21anc 857 . . 3 (𝜑 → ((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
175 f1oeq3 6354 . . 3 (((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))) → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
176174, 175syl 17 . 2 (𝜑 → ((𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((𝐻𝐼) +𝑜 ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
177122, 176mpbid 223 1 (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2989  wral 3107  Vcvv 3402  cun 3778  wss 3780  c0 4127   class class class wbr 4855  cmpt 4934   E cep 5236   We wwe 5282  ccnv 5323  dom cdm 5324  cima 5327  Ord word 5948  Oncon0 5949  suc csuc 5951  Fun wfun 6104   Fn wfn 6105  wf 6106  1-1-ontowf1o 6109  cfv 6110   Isom wiso 6111  (class class class)co 6883  cmpt2 6885  ωcom 7304   supp csupp 7538  seq𝜔cseqom 7787   +𝑜 coa 7802   ·𝑜 comu 7803  𝑜 coe 7804   finSupp cfsupp 8523  OrdIsocoi 8662   CNF ccnf 8814
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 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188  ax-inf2 8794
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 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-se 5284  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5906  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-isom 6119  df-riota 6844  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-om 7305  df-1st 7407  df-2nd 7408  df-supp 7539  df-wrecs 7651  df-recs 7713  df-rdg 7751  df-seqom 7788  df-1o 7805  df-2o 7806  df-oadd 7809  df-omul 7810  df-oexp 7811  df-er 7988  df-map 8103  df-en 8202  df-dom 8203  df-sdom 8204  df-fin 8205  df-fsupp 8524  df-oi 8663  df-cnf 8815
This theorem is referenced by:  cnfcom  8853
  Copyright terms: Public domain W3C validator