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Theorem naddcnfass 42700
Description: Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)
Assertion
Ref Expression
naddcnfass (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → ((𝐹f +o 𝐺) ∘f +o 𝐻) = (𝐹f +o (𝐺f +o 𝐻)))

Proof of Theorem naddcnfass
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
21eleq2d 2813 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆𝐹 ∈ dom (ω CNF 𝑋)))
3 eqid 2726 . . . . . . . 8 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4 omelon 9643 . . . . . . . . 9 ω ∈ On
54a1i 11 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6 simpl 482 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
73, 5, 6cantnfs 9663 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
82, 7bitrd 279 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
9 simpl 482 . . . . . . 7 ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹:𝑋⟶ω)
109ffnd 6712 . . . . . 6 ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹 Fn 𝑋)
118, 10syl6bi 253 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆𝐹 Fn 𝑋))
12 simp1 1133 . . . . 5 ((𝐹𝑆𝐺𝑆𝐻𝑆) → 𝐹𝑆)
1311, 12impel 505 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐹 Fn 𝑋)
141eleq2d 2813 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆𝐺 ∈ dom (ω CNF 𝑋)))
153, 5, 6cantnfs 9663 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ dom (ω CNF 𝑋) ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
1614, 15bitrd 279 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆 ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
17 simpl 482 . . . . . . 7 ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺:𝑋⟶ω)
1817ffnd 6712 . . . . . 6 ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺 Fn 𝑋)
1916, 18syl6bi 253 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆𝐺 Fn 𝑋))
20 simp2 1134 . . . . 5 ((𝐹𝑆𝐺𝑆𝐻𝑆) → 𝐺𝑆)
2119, 20impel 505 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐺 Fn 𝑋)
226adantr 480 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝑋 ∈ On)
23 inidm 4213 . . . 4 (𝑋𝑋) = 𝑋
2413, 21, 22, 22, 23offn 7680 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → (𝐹f +o 𝐺) Fn 𝑋)
251eleq2d 2813 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆𝐻 ∈ dom (ω CNF 𝑋)))
263, 5, 6cantnfs 9663 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ dom (ω CNF 𝑋) ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
2725, 26bitrd 279 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆 ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
28 simpl 482 . . . . . 6 ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻:𝑋⟶ω)
2928ffnd 6712 . . . . 5 ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻 Fn 𝑋)
3027, 29syl6bi 253 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆𝐻 Fn 𝑋))
31 simp3 1135 . . . 4 ((𝐹𝑆𝐺𝑆𝐻𝑆) → 𝐻𝑆)
3230, 31impel 505 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐻 Fn 𝑋)
3324, 32, 22, 22, 23offn 7680 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → ((𝐹f +o 𝐺) ∘f +o 𝐻) Fn 𝑋)
3421, 32, 22, 22, 23offn 7680 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → (𝐺f +o 𝐻) Fn 𝑋)
3513, 34, 22, 22, 23offn 7680 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → (𝐹f +o (𝐺f +o 𝐻)) Fn 𝑋)
368, 9syl6bi 253 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆𝐹:𝑋⟶ω))
3736, 12impel 505 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐹:𝑋⟶ω)
3837ffvelcdmda 7080 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ω)
3916, 17syl6bi 253 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆𝐺:𝑋⟶ω))
4039, 20impel 505 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐺:𝑋⟶ω)
4140ffvelcdmda 7080 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐺𝑥) ∈ ω)
4227, 28syl6bi 253 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆𝐻:𝑋⟶ω))
4342, 31impel 505 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐻:𝑋⟶ω)
4443ffvelcdmda 7080 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐻𝑥) ∈ ω)
45 nnaass 8623 . . . . 5 (((𝐹𝑥) ∈ ω ∧ (𝐺𝑥) ∈ ω ∧ (𝐻𝑥) ∈ ω) → (((𝐹𝑥) +o (𝐺𝑥)) +o (𝐻𝑥)) = ((𝐹𝑥) +o ((𝐺𝑥) +o (𝐻𝑥))))
4638, 41, 44, 45syl3anc 1368 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹𝑥) +o (𝐺𝑥)) +o (𝐻𝑥)) = ((𝐹𝑥) +o ((𝐺𝑥) +o (𝐻𝑥))))
4713adantr 480 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → 𝐹 Fn 𝑋)
4821adantr 480 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → 𝐺 Fn 𝑋)
4922anim1i 614 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝑋 ∈ On ∧ 𝑥𝑋))
50 fnfvof 7684 . . . . . 6 (((𝐹 Fn 𝑋𝐺 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝐹f +o 𝐺)‘𝑥) = ((𝐹𝑥) +o (𝐺𝑥)))
5147, 48, 49, 50syl21anc 835 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → ((𝐹f +o 𝐺)‘𝑥) = ((𝐹𝑥) +o (𝐺𝑥)))
5251oveq1d 7420 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹f +o 𝐺)‘𝑥) +o (𝐻𝑥)) = (((𝐹𝑥) +o (𝐺𝑥)) +o (𝐻𝑥)))
5332adantr 480 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → 𝐻 Fn 𝑋)
54 fnfvof 7684 . . . . . 6 (((𝐺 Fn 𝑋𝐻 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝐺f +o 𝐻)‘𝑥) = ((𝐺𝑥) +o (𝐻𝑥)))
5548, 53, 49, 54syl21anc 835 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → ((𝐺f +o 𝐻)‘𝑥) = ((𝐺𝑥) +o (𝐻𝑥)))
5655oveq2d 7421 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → ((𝐹𝑥) +o ((𝐺f +o 𝐻)‘𝑥)) = ((𝐹𝑥) +o ((𝐺𝑥) +o (𝐻𝑥))))
5746, 52, 563eqtr4d 2776 . . 3 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹f +o 𝐺)‘𝑥) +o (𝐻𝑥)) = ((𝐹𝑥) +o ((𝐺f +o 𝐻)‘𝑥)))
5824adantr 480 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐹f +o 𝐺) Fn 𝑋)
59 fnfvof 7684 . . . 4 ((((𝐹f +o 𝐺) Fn 𝑋𝐻 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → (((𝐹f +o 𝐺) ∘f +o 𝐻)‘𝑥) = (((𝐹f +o 𝐺)‘𝑥) +o (𝐻𝑥)))
6058, 53, 49, 59syl21anc 835 . . 3 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹f +o 𝐺) ∘f +o 𝐻)‘𝑥) = (((𝐹f +o 𝐺)‘𝑥) +o (𝐻𝑥)))
6134adantr 480 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐺f +o 𝐻) Fn 𝑋)
62 fnfvof 7684 . . . 4 (((𝐹 Fn 𝑋 ∧ (𝐺f +o 𝐻) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝐹f +o (𝐺f +o 𝐻))‘𝑥) = ((𝐹𝑥) +o ((𝐺f +o 𝐻)‘𝑥)))
6347, 61, 49, 62syl21anc 835 . . 3 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → ((𝐹f +o (𝐺f +o 𝐻))‘𝑥) = ((𝐹𝑥) +o ((𝐺f +o 𝐻)‘𝑥)))
6457, 60, 633eqtr4d 2776 . 2 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹f +o 𝐺) ∘f +o 𝐻)‘𝑥) = ((𝐹f +o (𝐺f +o 𝐻))‘𝑥))
6533, 35, 64eqfnfvd 7029 1 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → ((𝐹f +o 𝐺) ∘f +o 𝐻) = (𝐹f +o (𝐺f +o 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  c0 4317   class class class wbr 5141  dom cdm 5669  Oncon0 6358   Fn wfn 6532  wf 6533  cfv 6537  (class class class)co 7405  f cof 7665  ωcom 7852   +o coa 8464   finSupp cfsupp 9363   CNF ccnf 9658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seqom 8449  df-oadd 8471  df-map 8824  df-cnf 9659
This theorem is referenced by: (None)
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