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Theorem tfsconcatfv 43331
Description: The value of the concatenation of two transfinite series. (Contributed by RP, 24-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
tfsconcatfv ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋𝐶, (𝐴𝑋), (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑑,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑑,𝑥,𝑦,𝑧   𝐶,𝑎,𝑏,𝑑,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑑,𝑥,𝑦,𝑧   𝑋,𝑑,𝑥,𝑦,𝑧   + ,𝑑
Allowed substitution hints:   + (𝑥,𝑦,𝑧,𝑎,𝑏)   𝑋(𝑎,𝑏)

Proof of Theorem tfsconcatfv
StepHypRef Expression
1 tfsconcat.op . . . . 5 + = (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)))}))
21tfsconcatfv1 43329 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴𝑋))
32adantlr 715 . . 3 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴𝑋))
4 simpr 484 . . . 4 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋𝐶) → 𝑋𝐶)
54iftrued 4539 . . 3 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋𝐶) → if(𝑋𝐶, (𝐴𝑋), (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐴𝑋))
63, 5eqtr4d 2778 . 2 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋𝐶) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋𝐶, (𝐴𝑋), (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))))
7 simpr 484 . . . 4 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ¬ 𝑋𝐶)
87iffalsed 4542 . . 3 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → if(𝑋𝐶, (𝐴𝑋), (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)))
9 simpll 767 . . . 4 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)))
10 onss 7804 . . . . . . . 8 (𝐷 ∈ On → 𝐷 ⊆ On)
1110adantl 481 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ⊆ On)
1211ad3antlr 731 . . . . . 6 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → 𝐷 ⊆ On)
13 simpllr 776 . . . . . . 7 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
14 simplrl 777 . . . . . . . . 9 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝐶 ∈ On)
15 oacl 8572 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
1615adantl 481 . . . . . . . . . 10 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) ∈ On)
17 onelon 6411 . . . . . . . . . 10 (((𝐶 +o 𝐷) ∈ On ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
1816, 17sylan 580 . . . . . . . . 9 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
19 ontri1 6420 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶𝑋 ↔ ¬ 𝑋𝐶))
2014, 18, 19syl2anc 584 . . . . . . . 8 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → (𝐶𝑋 ↔ ¬ 𝑋𝐶))
2120biimpar 477 . . . . . . 7 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → 𝐶𝑋)
22 simplr 769 . . . . . . 7 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → 𝑋 ∈ (𝐶 +o 𝐷))
23 oawordex2 43316 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶𝑋𝑋 ∈ (𝐶 +o 𝐷))) → ∃𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)
2413, 21, 22, 23syl12anc 837 . . . . . 6 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ∃𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)
2514, 18jca 511 . . . . . . 7 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → (𝐶 ∈ On ∧ 𝑋 ∈ On))
26 oawordeu 8592 . . . . . . 7 (((𝐶 ∈ On ∧ 𝑋 ∈ On) ∧ 𝐶𝑋) → ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋)
2725, 21, 26syl2an2r 685 . . . . . 6 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋)
28 reuss 4333 . . . . . 6 ((𝐷 ⊆ On ∧ ∃𝑑𝐷 (𝐶 +o 𝑑) = 𝑋 ∧ ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋) → ∃!𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)
2912, 24, 27, 28syl3anc 1370 . . . . 5 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ∃!𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)
30 riotacl 7405 . . . . 5 (∃!𝑑𝐷 (𝐶 +o 𝑑) = 𝑋 → (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷)
3129, 30syl 17 . . . 4 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷)
321tfsconcatfv2 43330 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)))
339, 31, 32syl2anc 584 . . 3 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ((𝐴 + 𝐵)‘(𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)))
34 riotasbc 7406 . . . . . 6 (∃!𝑑𝐷 (𝐶 +o 𝑑) = 𝑋[(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋)
3529, 34syl 17 . . . . 5 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → [(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋)
36 sbceq1g 4423 . . . . . . 7 ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ([(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑(𝐶 +o 𝑑) = 𝑋))
37 csbov2g 7479 . . . . . . . . 9 ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑(𝐶 +o 𝑑) = (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑𝑑))
38 csbvarg 4440 . . . . . . . . . 10 ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑𝑑 = (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))
3938oveq2d 7447 . . . . . . . . 9 ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑𝑑) = (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)))
4037, 39eqtrd 2775 . . . . . . . 8 ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑(𝐶 +o 𝑑) = (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)))
4140eqeq1d 2737 . . . . . . 7 ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑(𝐶 +o 𝑑) = 𝑋 ↔ (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋))
4236, 41bitrd 279 . . . . . 6 ((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ([(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋 ↔ (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋))
4342biimpa 476 . . . . 5 (((𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷[(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋) → (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋)
4431, 35, 43syl2anc 584 . . . 4 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → (𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋)
4544fveq2d 6911 . . 3 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ((𝐴 + 𝐵)‘(𝐶 +o (𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))) = ((𝐴 + 𝐵)‘𝑋))
468, 33, 453eqtr2rd 2782 . 2 (((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋𝐶) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋𝐶, (𝐴𝑋), (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))))
476, 46pm2.61dan 813 1 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋𝐶, (𝐴𝑋), (𝐵‘(𝑑𝐷 (𝐶 +o 𝑑) = 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  ∃!wreu 3376  Vcvv 3478  [wsbc 3791  csb 3908  cdif 3960  cun 3961  wss 3963  ifcif 4531  {copab 5210  dom cdm 5689  Oncon0 6386   Fn wfn 6558  cfv 6563  crio 7387  (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-riota 7388  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)
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