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Theorem eceqoveq 8762
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 5671 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
21ad2antrr 725 . . . . . . 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 486 . . . . . . . 8 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] )
64, 5ereldm 8697 . . . . . . 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 5676 . . . . . 6 (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) → 𝐷𝑆)
97, 8syl 17 . . . . 5 ((((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) ∧ [⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ) → 𝐷𝑆)
109ex 414 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] 𝐷𝑆))
11 eceqoveq.9 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
1211caovcl 7549 . . . . . . 7 ((𝐵𝑆𝐶𝑆) → (𝐵 + 𝐶) ∈ 𝑆)
13 eleq1 2826 . . . . . . 7 ((𝐴 + 𝐷) = (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝑆 ↔ (𝐵 + 𝐶) ∈ 𝑆))
1412, 13syl5ibr 246 . . . . . 6 ((𝐴 + 𝐷) = (𝐵 + 𝐶) → ((𝐵𝑆𝐶𝑆) → (𝐴 + 𝐷) ∈ 𝑆))
15 eceqoveq.7 . . . . . . . 8 dom + = (𝑆 × 𝑆)
16 eceqoveq.8 . . . . . . . 8 ¬ ∅ ∈ 𝑆
1715, 16ndmovrcl 7541 . . . . . . 7 ((𝐴 + 𝐷) ∈ 𝑆 → (𝐴𝑆𝐷𝑆))
1817simprd 497 . . . . . 6 ((𝐴 + 𝐷) ∈ 𝑆𝐷𝑆)
1914, 18syl6com 37 . . . . 5 ((𝐵𝑆𝐶𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) → 𝐷𝑆))
2019adantll 713 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) → 𝐷𝑆))
213a1i 11 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → Er (𝑆 × 𝑆))
221adantr 482 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2321, 22erth 8698 . . . . . 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 281 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
2625expr 458 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → (𝐷𝑆 → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))))
2710, 20, 26pm5.21ndd 381 . . 3 (((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
2827an32s 651 . 2 (((𝐴𝑆𝐶𝑆) ∧ 𝐵𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
29 eqcom 2744 . . . 4 (∅ = [⟨𝐶, 𝐷⟩] ↔ [⟨𝐶, 𝐷⟩] = ∅)
30 erdm 8659 . . . . . . . . . . . 12 ( Er (𝑆 × 𝑆) → dom = (𝑆 × 𝑆))
313, 30ax-mp 5 . . . . . . . . . . 11 dom = (𝑆 × 𝑆)
3231eleq2i 2830 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ dom ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
33 ecdmn0 8696 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ dom ↔ [⟨𝐶, 𝐷⟩] ≠ ∅)
34 opelxp 5670 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) ↔ (𝐶𝑆𝐷𝑆))
3532, 33, 343bitr3i 301 . . . . . . . . 9 ([⟨𝐶, 𝐷⟩] ≠ ∅ ↔ (𝐶𝑆𝐷𝑆))
3635simplbi2 502 . . . . . . . 8 (𝐶𝑆 → (𝐷𝑆 → [⟨𝐶, 𝐷⟩] ≠ ∅))
3736ad2antlr 726 . . . . . . 7 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (𝐷𝑆 → [⟨𝐶, 𝐷⟩] ≠ ∅))
3837necon2bd 2960 . . . . . 6 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐶, 𝐷⟩] = ∅ → ¬ 𝐷𝑆))
39 simpr 486 . . . . . . 7 ((𝐴𝑆𝐷𝑆) → 𝐷𝑆)
4015ndmov 7539 . . . . . . 7 (¬ (𝐴𝑆𝐷𝑆) → (𝐴 + 𝐷) = ∅)
4139, 40nsyl5 159 . . . . . 6 𝐷𝑆 → (𝐴 + 𝐷) = ∅)
4238, 41syl6 35 . . . . 5 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐶, 𝐷⟩] = ∅ → (𝐴 + 𝐷) = ∅))
43 eleq1 2826 . . . . . . 7 ((𝐴 + 𝐷) = ∅ → ((𝐴 + 𝐷) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
4416, 43mtbiri 327 . . . . . 6 ((𝐴 + 𝐷) = ∅ → ¬ (𝐴 + 𝐷) ∈ 𝑆)
4535simprbi 498 . . . . . . . 8 ([⟨𝐶, 𝐷⟩] ≠ ∅ → 𝐷𝑆)
4611caovcl 7549 . . . . . . . . . 10 ((𝐴𝑆𝐷𝑆) → (𝐴 + 𝐷) ∈ 𝑆)
4746ex 414 . . . . . . . . 9 (𝐴𝑆 → (𝐷𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
4847ad2antrr 725 . . . . . . . 8 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (𝐷𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
4945, 48syl5 34 . . . . . . 7 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐶, 𝐷⟩] ≠ ∅ → (𝐴 + 𝐷) ∈ 𝑆))
5049necon1bd 2962 . . . . . 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 283 . . 3 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (∅ = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = ∅))
5431eleq2i 2830 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
55 ecdmn0 8696 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom ↔ [⟨𝐴, 𝐵⟩] ≠ ∅)
56 opelxp 5670 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
5754, 55, 563bitr3i 301 . . . . . . 7 ([⟨𝐴, 𝐵⟩] ≠ ∅ ↔ (𝐴𝑆𝐵𝑆))
5857simprbi 498 . . . . . 6 ([⟨𝐴, 𝐵⟩] ≠ ∅ → 𝐵𝑆)
5958necon1bi 2973 . . . . 5 𝐵𝑆 → [⟨𝐴, 𝐵⟩] = ∅)
6059adantl 483 . . . 4 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → [⟨𝐴, 𝐵⟩] = ∅)
6160eqeq1d 2739 . . 3 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ ∅ = [⟨𝐶, 𝐷⟩] ))
62 simpl 484 . . . . . 6 ((𝐵𝑆𝐶𝑆) → 𝐵𝑆)
6315ndmov 7539 . . . . . 6 (¬ (𝐵𝑆𝐶𝑆) → (𝐵 + 𝐶) = ∅)
6462, 63nsyl5 159 . . . . 5 𝐵𝑆 → (𝐵 + 𝐶) = ∅)
6564adantl 483 . . . 4 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → (𝐵 + 𝐶) = ∅)
6665eqeq2d 2748 . . 3 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐷) = ∅))
6753, 61, 663bitr4d 311 . 2 (((𝐴𝑆𝐶𝑆) ∧ ¬ 𝐵𝑆) → ([⟨𝐴, 𝐵⟩] = [⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
6828, 67pm2.61dan 812 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 397   = wceq 1542  wcel 2107  wne 2944  c0 4283  cop 4593   class class class wbr 5106   × cxp 5632  dom cdm 5634  (class class class)co 7358   Er wer 8646  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fv 6505  df-ov 7361  df-er 8649  df-ec 8651
This theorem is referenced by: (None)
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