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Theorem ditgeq123i 36188
Description: Equality inference for the directed integral. General version of ditgeq12i 36189 and ditgeq3i 36190. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
ditgeq123i.1 𝐴 = 𝐵
ditgeq123i.2 𝐶 = 𝐷
ditgeq123i.3 𝐸 = 𝐹
Assertion
Ref Expression
ditgeq123i ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐹 d𝑥

Proof of Theorem ditgeq123i
StepHypRef Expression
1 ditgeq123i.1 . . . 4 𝐴 = 𝐵
2 ditgeq123i.2 . . . 4 𝐶 = 𝐷
31, 2breq12i 5150 . . 3 (𝐴𝐶𝐵𝐷)
41, 2oveq12i 7441 . . . 4 (𝐴(,)𝐶) = (𝐵(,)𝐷)
5 ditgeq123i.3 . . . 4 𝐸 = 𝐹
64, 5itgeq12i 36185 . . 3 ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑥
72, 1oveq12i 7441 . . . . 5 (𝐶(,)𝐴) = (𝐷(,)𝐵)
87, 5itgeq12i 36185 . . . 4 ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑥
98negeqi 11497 . . 3 -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑥
103, 6, 9ifbieq12i 4551 . 2 if(𝐴𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥) = if(𝐵𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑥, -∫(𝐷(,)𝐵)𝐹 d𝑥)
11 df-ditg 25872 . 2 ⨜[𝐴𝐶]𝐸 d𝑥 = if(𝐴𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
12 df-ditg 25872 . 2 ⨜[𝐵𝐷]𝐹 d𝑥 = if(𝐵𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑥, -∫(𝐷(,)𝐵)𝐹 d𝑥)
1310, 11, 123eqtr4i 2774 1 ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐹 d𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  ifcif 4524   class class class wbr 5141  (class class class)co 7429  cle 11292  -cneg 11489  (,)cioo 13383  citg 25643  cdit 25871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  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-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-iota 6512  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-neg 11491  df-seq 14039  df-sum 15719  df-itg 25648  df-ditg 25872
This theorem is referenced by:  ditgeq12i  36189  ditgeq3i  36190
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