Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tfsconcatfn Structured version   Visualization version   GIF version

Theorem tfsconcatfn 43328
Description: The concatenation of two transfinite series is a transfinite series. (Contributed by RP, 22-Feb-2025.)
Hypothesis
Ref Expression
tfsconcat.op + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
Assertion
Ref Expression
tfsconcatfn (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧   𝐶,𝑎,𝑏,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑥,𝑦,𝑧
Allowed substitution hints:   + (𝑥,𝑦,𝑧,𝑎,𝑏)

Proof of Theorem tfsconcatfn
StepHypRef Expression
1 simpll 767 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐴 Fn 𝐶)
2 simplrl 777 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐶 ∈ On)
3 simplrr 778 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐷 ∈ On)
4 simpr 484 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
5 tfsconcatlem 43326 . . . . . 6 ((𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
62, 3, 4, 5syl3anc 1370 . . . . 5 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
76ralrimiva 3144 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∀𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
8 eqid 2735 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}
98fnopabg 6706 . . . 4 (∀𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
107, 9sylib 218 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
11 disjdif 4478 . . . 4 (𝐶 ∩ ((𝐶 +o 𝐷) ∖ 𝐶)) = ∅
1211a1i 11 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∩ ((𝐶 +o 𝐷) ∖ 𝐶)) = ∅)
131, 10, 12fnund 6684 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}) Fn (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)))
14 tfsconcat.op . . . 4 + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
1514tfsconcatun 43327 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}))
16 oaword1 8589 . . . . . 6 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷))
17 undif 4488 . . . . . 6 (𝐶 ⊆ (𝐶 +o 𝐷) ↔ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
1816, 17sylib 218 . . . . 5 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
1918eqcomd 2741 . . . 4 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) = (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)))
2019adantl 481 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) = (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)))
2115, 20fneq12d 6664 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 + 𝐵) Fn (𝐶 +o 𝐷) ↔ (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}) Fn (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))))
2213, 21mpbird 257 1 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ∃!weu 2566  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {copab 5210  dom cdm 5689  Oncon0 6386   Fn wfn 6558  cfv 6563  (class class class)co 7431  cmpo 7433   +o coa 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509
This theorem is referenced by:  tfsconcatfo  43333  tfsconcat0b  43336  tfsconcat00  43337
  Copyright terms: Public domain W3C validator