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Theorem tfsconcat0b 43364
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 7894 . . . . 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 43363 . . 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 5913 . . 3 ((𝐴 + 𝐵) = 𝐵 → dom (𝐴 + 𝐵) = dom 𝐵)
8 nna0r 8648 . . . . . . . . 9 (𝐷 ∈ ω → (∅ +o 𝐷) = 𝐷)
98adantl 481 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ +o 𝐷) = 𝐷)
109eqeq2d 2747 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (𝐶 +o 𝐷) = 𝐷))
11 eqcom 2743 . . . . . . 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 6439 . . . . . . . . . 10 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
1413adantr 480 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶𝐶 ≠ ∅))
15 df-ne 2940 . . . . . . . . 9 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
1614, 15bitr2di 288 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ ↔ ∅ ∈ 𝐶))
17 peano1 7911 . . . . . . . . . . . . . . 15 ∅ ∈ ω
18 nnaordr 8659 . . . . . . . . . . . . . . 15 ((∅ ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
1917, 18mp3an1 1449 . . . . . . . . . . . . . 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 8591 . . . . . . . . . . . . . . . . . 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 3993 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ (𝐶 +o 𝐷))
28 id 22 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ ω → 𝐷 ∈ ω)
298, 28eqeltrd 2840 . . . . . . . . . . . . . . . 16 (𝐷 ∈ ω → (∅ +o 𝐷) ∈ ω)
3029ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ ω)
3127, 30sseldd 3983 . . . . . . . . . . . . . 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 6393 . . . . . . . . . . . 12 (𝐶 ∈ On → Ord 𝐶)
36 ordom 7898 . . . . . . . . . . . 12 Ord ω
37 ordtri2or 6481 . . . . . . . . . . . 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 9639 . . . . . . . . . 10 ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → (∅ +o 𝐷) ≠ (𝐶 +o 𝐷))
4241neneqd 2944 . . . . . . . . 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 43356 . . . . . . 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 6672 . . . . 5 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom (𝐴 + 𝐵) = (𝐶 +o 𝐷))
51 fndm 6670 . . . . . 6 (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
5251ad2antlr 727 . . . . 5 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom 𝐵 = 𝐷)
5350, 52eqeq12d 2752 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 ↔ (𝐶 +o 𝐷) = 𝐷))
54 fnrel 6669 . . . . . . . 8 (𝐴 Fn 𝐶 → Rel 𝐴)
55 reldm0 5937 . . . . . . . 8 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
5654, 55syl 17 . . . . . . 7 (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
57 fndm 6670 . . . . . . . 8 (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
5857eqeq1d 2738 . . . . . . 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 1539  wcel 2107  wne 2939  wrex 3069  Vcvv 3479  cdif 3947  cun 3948  wss 3950  c0 4332  {copab 5204  dom cdm 5684  Rel wrel 5689  Ord word 6382  Oncon0 6383   Fn wfn 6555  cfv 6560  (class class class)co 7432  cmpo 7434  ωcom 7888   +o coa 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-oadd 8511
This theorem is referenced by: (None)
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