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Theorem naddcnfass 42863
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 483 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
21eleq2d 2811 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆𝐹 ∈ dom (ω CNF 𝑋)))
3 eqid 2725 . . . . . . . 8 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4 omelon 9669 . . . . . . . . 9 ω ∈ On
54a1i 11 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6 simpl 481 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
73, 5, 6cantnfs 9689 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
82, 7bitrd 278 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
9 simpl 481 . . . . . . 7 ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹:𝑋⟶ω)
109ffnd 6718 . . . . . 6 ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹 Fn 𝑋)
118, 10biimtrdi 252 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆𝐹 Fn 𝑋))
12 simp1 1133 . . . . 5 ((𝐹𝑆𝐺𝑆𝐻𝑆) → 𝐹𝑆)
1311, 12impel 504 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐹 Fn 𝑋)
141eleq2d 2811 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆𝐺 ∈ dom (ω CNF 𝑋)))
153, 5, 6cantnfs 9689 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ dom (ω CNF 𝑋) ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
1614, 15bitrd 278 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆 ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
17 simpl 481 . . . . . . 7 ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺:𝑋⟶ω)
1817ffnd 6718 . . . . . 6 ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺 Fn 𝑋)
1916, 18biimtrdi 252 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆𝐺 Fn 𝑋))
20 simp2 1134 . . . . 5 ((𝐹𝑆𝐺𝑆𝐻𝑆) → 𝐺𝑆)
2119, 20impel 504 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐺 Fn 𝑋)
226adantr 479 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝑋 ∈ On)
23 inidm 4213 . . . 4 (𝑋𝑋) = 𝑋
2413, 21, 22, 22, 23offn 7695 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → (𝐹f +o 𝐺) Fn 𝑋)
251eleq2d 2811 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆𝐻 ∈ dom (ω CNF 𝑋)))
263, 5, 6cantnfs 9689 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻 ∈ dom (ω CNF 𝑋) ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
2725, 26bitrd 278 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆 ↔ (𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅)))
28 simpl 481 . . . . . 6 ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻:𝑋⟶ω)
2928ffnd 6718 . . . . 5 ((𝐻:𝑋⟶ω ∧ 𝐻 finSupp ∅) → 𝐻 Fn 𝑋)
3027, 29biimtrdi 252 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆𝐻 Fn 𝑋))
31 simp3 1135 . . . 4 ((𝐹𝑆𝐺𝑆𝐻𝑆) → 𝐻𝑆)
3230, 31impel 504 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐻 Fn 𝑋)
3324, 32, 22, 22, 23offn 7695 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → ((𝐹f +o 𝐺) ∘f +o 𝐻) Fn 𝑋)
3421, 32, 22, 22, 23offn 7695 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → (𝐺f +o 𝐻) Fn 𝑋)
3513, 34, 22, 22, 23offn 7695 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → (𝐹f +o (𝐺f +o 𝐻)) Fn 𝑋)
368, 9biimtrdi 252 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆𝐹:𝑋⟶ω))
3736, 12impel 504 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐹:𝑋⟶ω)
3837ffvelcdmda 7089 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ω)
3916, 17biimtrdi 252 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺𝑆𝐺:𝑋⟶ω))
4039, 20impel 504 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐺:𝑋⟶ω)
4140ffvelcdmda 7089 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐺𝑥) ∈ ω)
4227, 28biimtrdi 252 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐻𝑆𝐻:𝑋⟶ω))
4342, 31impel 504 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) → 𝐻:𝑋⟶ω)
4443ffvelcdmda 7089 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐻𝑥) ∈ ω)
45 nnaass 8641 . . . . 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 479 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → 𝐹 Fn 𝑋)
4821adantr 479 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → 𝐺 Fn 𝑋)
4922anim1i 613 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝑋 ∈ On ∧ 𝑥𝑋))
50 fnfvof 7699 . . . . . 6 (((𝐹 Fn 𝑋𝐺 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝐹f +o 𝐺)‘𝑥) = ((𝐹𝑥) +o (𝐺𝑥)))
5147, 48, 49, 50syl21anc 836 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → ((𝐹f +o 𝐺)‘𝑥) = ((𝐹𝑥) +o (𝐺𝑥)))
5251oveq1d 7431 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹f +o 𝐺)‘𝑥) +o (𝐻𝑥)) = (((𝐹𝑥) +o (𝐺𝑥)) +o (𝐻𝑥)))
5332adantr 479 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → 𝐻 Fn 𝑋)
54 fnfvof 7699 . . . . . 6 (((𝐺 Fn 𝑋𝐻 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝐺f +o 𝐻)‘𝑥) = ((𝐺𝑥) +o (𝐻𝑥)))
5548, 53, 49, 54syl21anc 836 . . . . 5 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → ((𝐺f +o 𝐻)‘𝑥) = ((𝐺𝑥) +o (𝐻𝑥)))
5655oveq2d 7432 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → ((𝐹𝑥) +o ((𝐺f +o 𝐻)‘𝑥)) = ((𝐹𝑥) +o ((𝐺𝑥) +o (𝐻𝑥))))
5746, 52, 563eqtr4d 2775 . . 3 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹f +o 𝐺)‘𝑥) +o (𝐻𝑥)) = ((𝐹𝑥) +o ((𝐺f +o 𝐻)‘𝑥)))
5824adantr 479 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐹f +o 𝐺) Fn 𝑋)
59 fnfvof 7699 . . . 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 836 . . 3 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (((𝐹f +o 𝐺) ∘f +o 𝐻)‘𝑥) = (((𝐹f +o 𝐺)‘𝑥) +o (𝐻𝑥)))
6134adantr 479 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹𝑆𝐺𝑆𝐻𝑆)) ∧ 𝑥𝑋) → (𝐺f +o 𝐻) Fn 𝑋)
62 fnfvof 7699 . . . 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 836 . . 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 2775 . 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 7038 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 394  w3a 1084   = wceq 1533  wcel 2098  c0 4318   class class class wbr 5143  dom cdm 5672  Oncon0 6364   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7416  f cof 7680  ωcom 7868   +o coa 8482   finSupp cfsupp 9385   CNF ccnf 9684
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seqom 8467  df-oadd 8489  df-map 8845  df-cnf 9685
This theorem is referenced by: (None)
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