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Theorem eceqoveq 8851
Description: Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
eceqoveq.5 Er (𝑆 × 𝑆)
eceqoveq.7 dom + = (𝑆 × 𝑆)
eceqoveq.8 ¬ ∅ ∈ 𝑆
eceqoveq.9 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
eceqoveq.10 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
Assertion
Ref Expression
eceqoveq ((𝐴𝑆𝐶𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
Distinct variable groups:   𝑥,𝑦, +   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem eceqoveq
StepHypRef Expression
1 opelxpi 5719 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
21ad2antrr 724 . . . . . . 7 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
3 eceqoveq.5 . . . . . . . . 9 Er (𝑆 × 𝑆)
43a1i 11 . . . . . . . 8 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → Er (𝑆 × 𝑆))
5 simpr 483 . . . . . . . 8 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] )
64, 5ereldm 8784 . . . . . . 7 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
72, 6mpbid 231 . . . . . 6 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
8 opelxp2 5725 . . . . . 6 (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) → 𝐷𝑆)
97, 8syl 17 . . . . 5 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → 𝐷𝑆)
109ex 411 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] 𝐷𝑆))
11 eceqoveq.9 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
1211caovcl 7620 . . . . . . 7 ((𝐵𝑆𝐶𝑆) → (𝐵 + 𝐶) ∈ 𝑆)
13 eleq1 2814 . . . . . . 7 ((𝐴 + 𝐷) = (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝑆 ↔ (𝐵 + 𝐶) ∈ 𝑆))
1412, 13imbitrrid 245 . . . . . 6 ((𝐴 + 𝐷) = (𝐵 + 𝐶) → ((𝐵𝑆𝐶𝑆) → (𝐴 + 𝐷) ∈ 𝑆))
15 eceqoveq.7 . . . . . . . 8 dom + = (𝑆 × 𝑆)
16 eceqoveq.8 . . . . . . . 8 ¬ ∅ ∈ 𝑆
1715, 16ndmovrcl 7612 . . . . . . 7 ((𝐴 + 𝐷) ∈ 𝑆 → (𝐴𝑆𝐷𝑆))
1817simprd 494 . . . . . 6 ((𝐴 + 𝐷) ∈ 𝑆𝐷𝑆)
1914, 18syl6com 37 . . . . 5 ((𝐵𝑆𝐶𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) → 𝐷𝑆))
2019adantll 712 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) → 𝐷𝑆))
213a1i 11 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → Er (𝑆 × 𝑆))
221adantr 479 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2321, 22erth 8785 . . . . . 6 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ))
24 eceqoveq.10 . . . . . 6 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
2523, 24bitr3d 280 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
2625expr 455 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → (𝐷𝑆 → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))))
2710, 20, 26pm5.21ndd 378 . . 3 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
2827an32s 650 . 2 (((𝐴𝑆𝐶𝑆) ∧ 𝐵𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
29 eqcom 2733 . . . 4 (∅ = [⟨𝐶, 𝐷⟩] ↔ [⟨𝐶, 𝐷⟩] = ∅)
30 erdm 8744 . . . . . . . . . . . 12 ( Er (𝑆 × 𝑆) → dom = (𝑆 × 𝑆))
313, 30ax-mp 5 . . . . . . . . . . 11 dom = (𝑆 × 𝑆)
3231eleq2i 2818 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ dom ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
33 ecdmn0 8783 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ dom ↔ [⟨𝐶, 𝐷⟩] ≠ ∅)
34 opelxp 5718 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) ↔ (𝐶𝑆𝐷𝑆))
3532, 33, 343bitr3i 300 . . . . . . . . 9 ([⟨𝐶, 𝐷⟩] ≠ ∅ ↔ (𝐶𝑆𝐷𝑆))
3635simplbi2 499 . . . . . . . 8 (𝐶𝑆 → (𝐷𝑆 → [⟨𝐶, 𝐷⟩] ≠ ∅))
3736ad2antlr 725 . . . . . . 7 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (𝐷𝑆 → [⟨𝐶, 𝐷⟩] ≠ ∅))
3837necon2bd 2946 . . . . . 6 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐶, 𝐷⟩] = ∅ → ¬ 𝐷𝑆))
39 simpr 483 . . . . . . 7 ((𝐴𝑆𝐷𝑆) → 𝐷𝑆)
4015ndmov 7610 . . . . . . 7 (¬ (𝐴𝑆𝐷𝑆) → (𝐴 + 𝐷) = ∅)
4139, 40nsyl5 159 . . . . . 6 𝐷𝑆 → (𝐴 + 𝐷) = ∅)
4238, 41syl6 35 . . . . 5 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐶, 𝐷⟩] = ∅ → (𝐴 + 𝐷) = ∅))
43 eleq1 2814 . . . . . . 7 ((𝐴 + 𝐷) = ∅ → ((𝐴 + 𝐷) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
4416, 43mtbiri 326 . . . . . 6 ((𝐴 + 𝐷) = ∅ → ¬ (𝐴 + 𝐷) ∈ 𝑆)
4535simprbi 495 . . . . . . . 8 ([⟨𝐶, 𝐷⟩] ≠ ∅ → 𝐷𝑆)
4611caovcl 7620 . . . . . . . . . 10 ((𝐴𝑆𝐷𝑆) → (𝐴 + 𝐷) ∈ 𝑆)
4746ex 411 . . . . . . . . 9 (𝐴𝑆 → (𝐷𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
4847ad2antrr 724 . . . . . . . 8 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (𝐷𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
4945, 48syl5 34 . . . . . . 7 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐶, 𝐷⟩] ≠ ∅ → (𝐴 + 𝐷) ∈ 𝑆))
5049necon1bd 2948 . . . . . 6 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (¬ (𝐴 + 𝐷) ∈ 𝑆 → [⟨𝐶, 𝐷⟩] = ∅))
5144, 50syl5 34 . . . . 5 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ((𝐴 + 𝐷) = ∅ → [⟨𝐶, 𝐷⟩] = ∅))
5242, 51impbid 211 . . . 4 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐶, 𝐷⟩] = ∅ ↔ (𝐴 + 𝐷) = ∅))
5329, 52bitrid 282 . . 3 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (∅ = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = ∅))
5431eleq2i 2818 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
55 ecdmn0 8783 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom ↔ [⟨𝐴, 𝐵⟩] ≠ ∅)
56 opelxp 5718 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
5754, 55, 563bitr3i 300 . . . . . . 7 ([⟨𝐴, 𝐵⟩] ≠ ∅ ↔ (𝐴𝑆𝐵𝑆))
5857simprbi 495 . . . . . 6 ([⟨𝐴, 𝐵⟩] ≠ ∅ → 𝐵𝑆)
5958necon1bi 2959 . . . . 5 𝐵𝑆 → [⟨𝐴, 𝐵⟩] = ∅)
6059adantl 480 . . . 4 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → [⟨𝐴, 𝐵⟩] = ∅)
6160eqeq1d 2728 . . 3 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ ∅ = [⟨𝐶, 𝐷⟩] ))
62 simpl 481 . . . . . 6 ((𝐵𝑆𝐶𝑆) → 𝐵𝑆)
6315ndmov 7610 . . . . . 6 (¬ (𝐵𝑆𝐶𝑆) → (𝐵 + 𝐶) = ∅)
6462, 63nsyl5 159 . . . . 5 𝐵𝑆 → (𝐵 + 𝐶) = ∅)
6564adantl 480 . . . 4 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (𝐵 + 𝐶) = ∅)
6665eqeq2d 2737 . . 3 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐷) = ∅))
6753, 61, 663bitr4d 310 . 2 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
6828, 67pm2.61dan 811 1 ((𝐴𝑆𝐶𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  c0 4325  cop 4639   class class class wbr 5153   × cxp 5680  dom cdm 5682  (class class class)co 7424   Er wer 8731  [cec 8732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fv 6562  df-ov 7427  df-er 8734  df-ec 8736
This theorem is referenced by: (None)
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