Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tfsconcat0b Structured version   Visualization version   GIF version

Theorem tfsconcat0b 43336
Description: The concatentation with the empty series leaves the finite series unchanged. (Contributed by RP, 1-Mar-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
tfsconcat0b (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))
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 tfsconcat0b
StepHypRef Expression
1 nnon 7893 . . . . 5 (𝐷 ∈ ω → 𝐷 ∈ On)
21anim2i 617 . . . 4 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
32anim2i 617 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)))
4 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)))}))
54tfsconcat0i 43335 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))
63, 5syl 17 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))
7 dmeq 5917 . . 3 ((𝐴 + 𝐵) = 𝐵 → dom (𝐴 + 𝐵) = dom 𝐵)
8 nna0r 8646 . . . . . . . . 9 (𝐷 ∈ ω → (∅ +o 𝐷) = 𝐷)
98adantl 481 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ +o 𝐷) = 𝐷)
109eqeq2d 2746 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (𝐶 +o 𝐷) = 𝐷))
11 eqcom 2742 . . . . . . 7 ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷))
1210, 11bitr3di 286 . . . . . 6 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷 ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷)))
13 on0eln0 6442 . . . . . . . . . 10 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
1413adantr 480 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶𝐶 ≠ ∅))
15 df-ne 2939 . . . . . . . . 9 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
1614, 15bitr2di 288 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ ↔ ∅ ∈ 𝐶))
17 peano1 7911 . . . . . . . . . . . . . . 15 ∅ ∈ ω
18 nnaordr 8657 . . . . . . . . . . . . . . 15 ((∅ ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
1917, 18mp3an1 1447 . . . . . . . . . . . . . 14 ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
2019biimpd 229 . . . . . . . . . . . . 13 ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
2120ex 412 . . . . . . . . . . . 12 (𝐶 ∈ ω → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))))
2221a1i 11 . . . . . . . . . . 11 (𝐶 ∈ On → (𝐶 ∈ ω → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))))
23 simpr 484 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ 𝐶)
24 oaword1 8589 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷))
252, 24syl 17 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → 𝐶 ⊆ (𝐶 +o 𝐷))
2625adantr 480 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → 𝐶 ⊆ (𝐶 +o 𝐷))
2723, 26sstrd 4006 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ (𝐶 +o 𝐷))
28 id 22 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ ω → 𝐷 ∈ ω)
298, 28eqeltrd 2839 . . . . . . . . . . . . . . . 16 (𝐷 ∈ ω → (∅ +o 𝐷) ∈ ω)
3029ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ ω)
3127, 30sseldd 3996 . . . . . . . . . . . . . 14 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))
3231a1d 25 . . . . . . . . . . . . 13 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
3332exp31 419 . . . . . . . . . . . 12 (𝐶 ∈ On → (𝐷 ∈ ω → (ω ⊆ 𝐶 → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))))
3433com23 86 . . . . . . . . . . 11 (𝐶 ∈ On → (ω ⊆ 𝐶 → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))))
35 eloni 6396 . . . . . . . . . . . 12 (𝐶 ∈ On → Ord 𝐶)
36 ordom 7897 . . . . . . . . . . . 12 Ord ω
37 ordtri2or 6484 . . . . . . . . . . . 12 ((Ord 𝐶 ∧ Ord ω) → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
3835, 36, 37sylancl 586 . . . . . . . . . . 11 (𝐶 ∈ On → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
3922, 34, 38mpjaod 860 . . . . . . . . . 10 (𝐶 ∈ On → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))))
4039imp 406 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
41 elneq 9636 . . . . . . . . . 10 ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → (∅ +o 𝐷) ≠ (𝐶 +o 𝐷))
4241neneqd 2943 . . . . . . . . 9 ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → ¬ (∅ +o 𝐷) = (𝐶 +o 𝐷))
4340, 42syl6 35 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → ¬ (∅ +o 𝐷) = (𝐶 +o 𝐷)))
4416, 43sylbid 240 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ → ¬ (∅ +o 𝐷) = (𝐶 +o 𝐷)))
4544con4d 115 . . . . . 6 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((∅ +o 𝐷) = (𝐶 +o 𝐷) → 𝐶 = ∅))
4612, 45sylbid 240 . . . . 5 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷𝐶 = ∅))
4746adantl 481 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐶 +o 𝐷) = 𝐷𝐶 = ∅))
484tfsconcatfn 43328 . . . . . . 7 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
493, 48syl 17 . . . . . 6 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
5049fndmd 6674 . . . . 5 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom (𝐴 + 𝐵) = (𝐶 +o 𝐷))
51 fndm 6672 . . . . . 6 (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
5251ad2antlr 727 . . . . 5 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom 𝐵 = 𝐷)
5350, 52eqeq12d 2751 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 ↔ (𝐶 +o 𝐷) = 𝐷))
54 fnrel 6671 . . . . . . . 8 (𝐴 Fn 𝐶 → Rel 𝐴)
55 reldm0 5941 . . . . . . . 8 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
5654, 55syl 17 . . . . . . 7 (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
57 fndm 6672 . . . . . . . 8 (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
5857eqeq1d 2737 . . . . . . 7 (𝐴 Fn 𝐶 → (dom 𝐴 = ∅ ↔ 𝐶 = ∅))
5956, 58bitrd 279 . . . . . 6 (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ 𝐶 = ∅))
6059adantr 480 . . . . 5 ((𝐴 Fn 𝐶𝐵 Fn 𝐷) → (𝐴 = ∅ ↔ 𝐶 = ∅))
6160adantr 480 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ 𝐶 = ∅))
6247, 53, 613imtr4d 294 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵𝐴 = ∅))
637, 62syl5 34 . 2 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐴 + 𝐵) = 𝐵𝐴 = ∅))
646, 63impbid 212 1 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  wss 3963  c0 4339  {copab 5210  dom cdm 5689  Rel wrel 5694  Ord word 6385  Oncon0 6386   Fn wfn 6558  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887   +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  ax-reg 9630
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: (None)
  Copyright terms: Public domain W3C validator