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Theorem cantnflem3 8548
Description: Lemma for cantnf 8550. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 7643 to factor 𝐶 into the form ((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴𝑜 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴𝑜 𝑋) ≤ (𝐴𝑜 𝑋) ·𝑜 𝑌𝐶, 𝑍 has a normal form, and by appending the term (𝐴𝑜 𝑋) ·𝑜 𝑌 using cantnfp1 8538 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)
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)))}
cantnf.c (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
cantnf.s (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e (𝜑 → ∅ ∈ 𝐶)
cantnf.x 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}
cantnf.p 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))
cantnf.y 𝑌 = (1st𝑃)
cantnf.z 𝑍 = (2nd𝑃)
cantnf.g (𝜑𝐺𝑆)
cantnf.v (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
cantnf.f 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
Assertion
Ref Expression
cantnflem3 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Distinct variable groups:   𝑡,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧,𝐶   𝑡,𝑎,𝐴,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑡   𝑤,𝐹,𝑥,𝑦,𝑧   𝑆,𝑐,𝑡,𝑥,𝑦,𝑧   𝑡,𝑍,𝑥,𝑦,𝑧   𝐺,𝑐,𝑡,𝑤,𝑥,𝑦,𝑧   𝜑,𝑡,𝑥,𝑦,𝑧   𝑡,𝑌,𝑤,𝑥,𝑦,𝑧   𝑋,𝑎,𝑏,𝑑,𝑡,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑑)   𝐶(𝑡)   𝑃(𝑥,𝑦,𝑧,𝑤,𝑡,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑤,𝑎,𝑏,𝑑)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑑)   𝐹(𝑡,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑎,𝑏,𝑑)   𝑋(𝑐)   𝑌(𝑎,𝑏,𝑐,𝑑)   𝑍(𝑤,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem cantnflem3
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 cantnfs.s . . . . 5 𝑆 = dom (𝐴 CNF 𝐵)
2 cantnfs.a . . . . 5 (𝜑𝐴 ∈ On)
3 cantnfs.b . . . . 5 (𝜑𝐵 ∈ On)
4 cantnf.g . . . . 5 (𝜑𝐺𝑆)
5 oemapval.t . . . . . . . . . . . . . 14 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
6 cantnf.c . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
7 cantnf.s . . . . . . . . . . . . . 14 (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
8 cantnf.e . . . . . . . . . . . . . 14 (𝜑 → ∅ ∈ 𝐶)
91, 2, 3, 5, 6, 7, 8cantnflem2 8547 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
10 eqid 2621 . . . . . . . . . . . . . . 15 𝑋 = 𝑋
11 eqid 2621 . . . . . . . . . . . . . . 15 𝑌 = 𝑌
12 eqid 2621 . . . . . . . . . . . . . . 15 𝑍 = 𝑍
1310, 11, 123pm3.2i 1237 . . . . . . . . . . . . . 14 (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)
14 cantnf.x . . . . . . . . . . . . . . 15 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}
15 cantnf.p . . . . . . . . . . . . . . 15 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))
16 cantnf.y . . . . . . . . . . . . . . 15 𝑌 = (1st𝑃)
17 cantnf.z . . . . . . . . . . . . . . 15 𝑍 = (2nd𝑃)
1814, 15, 16, 17oeeui 7642 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶) ↔ (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)))
1913, 18mpbiri 248 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶))
209, 19syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶))
2120simpld 475 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)))
2221simp1d 1071 . . . . . . . . . 10 (𝜑𝑋 ∈ On)
23 oecl 7577 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
242, 22, 23syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
2521simp2d 1072 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐴 ∖ 1𝑜))
2625eldifad 3572 . . . . . . . . . 10 (𝜑𝑌𝐴)
27 onelon 5717 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
282, 26, 27syl2anc 692 . . . . . . . . 9 (𝜑𝑌 ∈ On)
29 dif1o 7540 . . . . . . . . . . . 12 (𝑌 ∈ (𝐴 ∖ 1𝑜) ↔ (𝑌𝐴𝑌 ≠ ∅))
3029simprbi 480 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1𝑜) → 𝑌 ≠ ∅)
3125, 30syl 17 . . . . . . . . . 10 (𝜑𝑌 ≠ ∅)
32 on0eln0 5749 . . . . . . . . . . 11 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3328, 32syl 17 . . . . . . . . . 10 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
3431, 33mpbird 247 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝑌)
35 omword1 7613 . . . . . . . . 9 ((((𝐴𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴𝑜 𝑋) ⊆ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
3624, 28, 34, 35syl21anc 1322 . . . . . . . 8 (𝜑 → (𝐴𝑜 𝑋) ⊆ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
37 omcl 7576 . . . . . . . . . . 11 (((𝐴𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On)
3824, 28, 37syl2anc 692 . . . . . . . . . 10 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On)
3921simp3d 1073 . . . . . . . . . . 11 (𝜑𝑍 ∈ (𝐴𝑜 𝑋))
40 onelon 5717 . . . . . . . . . . 11 (((𝐴𝑜 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) → 𝑍 ∈ On)
4124, 39, 40syl2anc 692 . . . . . . . . . 10 (𝜑𝑍 ∈ On)
42 oaword1 7592 . . . . . . . . . 10 ((((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ⊆ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
4338, 41, 42syl2anc 692 . . . . . . . . 9 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ⊆ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
4420simprd 479 . . . . . . . . 9 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶)
4543, 44sseqtrd 3626 . . . . . . . 8 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ⊆ 𝐶)
4636, 45sstrd 3598 . . . . . . 7 (𝜑 → (𝐴𝑜 𝑋) ⊆ 𝐶)
47 oecl 7577 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
482, 3, 47syl2anc 692 . . . . . . . 8 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
49 ontr2 5741 . . . . . . . 8 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → (((𝐴𝑜 𝑋) ⊆ 𝐶𝐶 ∈ (𝐴𝑜 𝐵)) → (𝐴𝑜 𝑋) ∈ (𝐴𝑜 𝐵)))
5024, 48, 49syl2anc 692 . . . . . . 7 (𝜑 → (((𝐴𝑜 𝑋) ⊆ 𝐶𝐶 ∈ (𝐴𝑜 𝐵)) → (𝐴𝑜 𝑋) ∈ (𝐴𝑜 𝐵)))
5146, 6, 50mp2and 714 . . . . . 6 (𝜑 → (𝐴𝑜 𝑋) ∈ (𝐴𝑜 𝐵))
529simpld 475 . . . . . . 7 (𝜑𝐴 ∈ (On ∖ 2𝑜))
53 oeord 7628 . . . . . . 7 ((𝑋 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑋𝐵 ↔ (𝐴𝑜 𝑋) ∈ (𝐴𝑜 𝐵)))
5422, 3, 52, 53syl3anc 1323 . . . . . 6 (𝜑 → (𝑋𝐵 ↔ (𝐴𝑜 𝑋) ∈ (𝐴𝑜 𝐵)))
5551, 54mpbird 247 . . . . 5 (𝜑𝑋𝐵)
562adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ On)
573adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐵 ∈ On)
58 suppssdm 7268 . . . . . . . . . . . . . . 15 (𝐺 supp ∅) ⊆ dom 𝐺
591, 2, 3cantnfs 8523 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
604, 59mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6160simpld 475 . . . . . . . . . . . . . . . 16 (𝜑𝐺:𝐵𝐴)
62 fdm 6018 . . . . . . . . . . . . . . . 16 (𝐺:𝐵𝐴 → dom 𝐺 = 𝐵)
6361, 62syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐺 = 𝐵)
6458, 63syl5sseq 3638 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6564sselda 3588 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝐵)
66 onelon 5717 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
6757, 65, 66syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ On)
68 oecl 7577 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑜 𝑥) ∈ On)
6956, 67, 68syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴𝑜 𝑥) ∈ On)
7061adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺:𝐵𝐴)
7170, 65ffvelrnd 6326 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ 𝐴)
72 onelon 5717 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐺𝑥) ∈ 𝐴) → (𝐺𝑥) ∈ On)
7356, 71, 72syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ On)
74 ffn 6012 . . . . . . . . . . . . . . 15 (𝐺:𝐵𝐴𝐺 Fn 𝐵)
7561, 74syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺 Fn 𝐵)
76 0ex 4760 . . . . . . . . . . . . . . 15 ∅ ∈ V
7776a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ∅ ∈ V)
78 elsuppfn 7263 . . . . . . . . . . . . . 14 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
7975, 3, 77, 78syl3anc 1323 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
8079simplbda 653 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ≠ ∅)
81 on0eln0 5749 . . . . . . . . . . . . 13 ((𝐺𝑥) ∈ On → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
8273, 81syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
8380, 82mpbird 247 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ∅ ∈ (𝐺𝑥))
84 omword1 7613 . . . . . . . . . . 11 ((((𝐴𝑜 𝑥) ∈ On ∧ (𝐺𝑥) ∈ On) ∧ ∅ ∈ (𝐺𝑥)) → (𝐴𝑜 𝑥) ⊆ ((𝐴𝑜 𝑥) ·𝑜 (𝐺𝑥)))
8569, 73, 83, 84syl21anc 1322 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴𝑜 𝑥) ⊆ ((𝐴𝑜 𝑥) ·𝑜 (𝐺𝑥)))
86 eqid 2621 . . . . . . . . . . . 12 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
874adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺𝑆)
88 eqid 2621 . . . . . . . . . . . 12 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)
891, 56, 57, 86, 87, 88, 65cantnfle 8528 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴𝑜 𝑥) ·𝑜 (𝐺𝑥)) ⊆ ((𝐴 CNF 𝐵)‘𝐺))
90 cantnf.v . . . . . . . . . . . 12 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
9190adantr 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
9289, 91sseqtrd 3626 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴𝑜 𝑥) ·𝑜 (𝐺𝑥)) ⊆ 𝑍)
9385, 92sstrd 3598 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴𝑜 𝑥) ⊆ 𝑍)
9439adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑍 ∈ (𝐴𝑜 𝑋))
9524adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴𝑜 𝑋) ∈ On)
96 ontr2 5741 . . . . . . . . . 10 (((𝐴𝑜 𝑥) ∈ On ∧ (𝐴𝑜 𝑋) ∈ On) → (((𝐴𝑜 𝑥) ⊆ 𝑍𝑍 ∈ (𝐴𝑜 𝑋)) → (𝐴𝑜 𝑥) ∈ (𝐴𝑜 𝑋)))
9769, 95, 96syl2anc 692 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (((𝐴𝑜 𝑥) ⊆ 𝑍𝑍 ∈ (𝐴𝑜 𝑋)) → (𝐴𝑜 𝑥) ∈ (𝐴𝑜 𝑋)))
9893, 94, 97mp2and 714 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴𝑜 𝑥) ∈ (𝐴𝑜 𝑋))
9922adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑋 ∈ On)
10052adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ (On ∖ 2𝑜))
101 oeord 7628 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑥𝑋 ↔ (𝐴𝑜 𝑥) ∈ (𝐴𝑜 𝑋)))
10267, 99, 100, 101syl3anc 1323 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝑥𝑋 ↔ (𝐴𝑜 𝑥) ∈ (𝐴𝑜 𝑋)))
10398, 102mpbird 247 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝑋)
104103ex 450 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) → 𝑥𝑋))
105104ssrdv 3594 . . . . 5 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
106 cantnf.f . . . . 5 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
1071, 2, 3, 4, 55, 26, 105, 106cantnfp1 8538 . . . 4 (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
108107simprd 479 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
10990oveq2d 6631 . . 3 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
110108, 109, 443eqtrd 2659 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = 𝐶)
1111, 2, 3cantnff 8531 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
112 ffn 6012 . . . 4 ((𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
113111, 112syl 17 . . 3 (𝜑 → (𝐴 CNF 𝐵) Fn 𝑆)
114107simpld 475 . . 3 (𝜑𝐹𝑆)
115 fnfvelrn 6322 . . 3 (((𝐴 CNF 𝐵) Fn 𝑆𝐹𝑆) → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
116113, 114, 115syl2anc 692 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
117110, 116eqeltrrd 2699 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  cdif 3557  wss 3560  c0 3897  ifcif 4064  cop 4161   cuni 4409   cint 4447   class class class wbr 4623  {copab 4682  cmpt 4683   E cep 4993  dom cdm 5084  ran crn 5085  Oncon0 5692  cio 5818   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617  1st c1st 7126  2nd c2nd 7127   supp csupp 7255  seq𝜔cseqom 7502  1𝑜c1o 7513  2𝑜c2o 7514   +𝑜 coa 7517   ·𝑜 comu 7518  𝑜 coe 7519   finSupp cfsupp 8235  OrdIsocoi 8374   CNF ccnf 8518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-seqom 7503  df-1o 7520  df-2o 7521  df-oadd 7524  df-omul 7525  df-oexp 7526  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-oi 8375  df-cnf 8519
This theorem is referenced by:  cantnflem4  8549
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