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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 ((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴 ↑𝑜 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴 ↑𝑜 𝑋) ≤ (𝐴 ↑𝑜 𝑋) ·𝑜 𝑌 ≤ 𝐶, 𝑍 has a normal form, and by appending the term (𝐴 ↑𝑜 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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