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Theorem addcmpblnr 10092
Description: Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addcmpblnr ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩))

Proof of Theorem addcmpblnr
StepHypRef Expression
1 oveq12 6802 . 2 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆)) = ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅)))
2 addclpr 10042 . . . . . . . 8 ((𝐴P𝐹P) → (𝐴 +P 𝐹) ∈ P)
3 addclpr 10042 . . . . . . . 8 ((𝐵P𝐺P) → (𝐵 +P 𝐺) ∈ P)
42, 3anim12i 600 . . . . . . 7 (((𝐴P𝐹P) ∧ (𝐵P𝐺P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
54an4s 639 . . . . . 6 (((𝐴P𝐵P) ∧ (𝐹P𝐺P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
6 addclpr 10042 . . . . . . . 8 ((𝐶P𝑅P) → (𝐶 +P 𝑅) ∈ P)
7 addclpr 10042 . . . . . . . 8 ((𝐷P𝑆P) → (𝐷 +P 𝑆) ∈ P)
86, 7anim12i 600 . . . . . . 7 (((𝐶P𝑅P) ∧ (𝐷P𝑆P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
98an4s 639 . . . . . 6 (((𝐶P𝐷P) ∧ (𝑅P𝑆P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
105, 9anim12i 600 . . . . 5 ((((𝐴P𝐵P) ∧ (𝐹P𝐺P)) ∧ ((𝐶P𝐷P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P) ∧ ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P)))
1110an4s 639 . . . 4 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P) ∧ ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P)))
12 enrbreq 10087 . . . 4 ((((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P) ∧ ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P)) → (⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩ ↔ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅))))
1311, 12syl 17 . . 3 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩ ↔ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅))))
14 addcompr 10045 . . . . . . . 8 (𝐹 +P 𝐷) = (𝐷 +P 𝐹)
1514oveq1i 6803 . . . . . . 7 ((𝐹 +P 𝐷) +P 𝑆) = ((𝐷 +P 𝐹) +P 𝑆)
16 addasspr 10046 . . . . . . 7 ((𝐹 +P 𝐷) +P 𝑆) = (𝐹 +P (𝐷 +P 𝑆))
17 addasspr 10046 . . . . . . 7 ((𝐷 +P 𝐹) +P 𝑆) = (𝐷 +P (𝐹 +P 𝑆))
1815, 16, 173eqtr3i 2801 . . . . . 6 (𝐹 +P (𝐷 +P 𝑆)) = (𝐷 +P (𝐹 +P 𝑆))
1918oveq2i 6804 . . . . 5 (𝐴 +P (𝐹 +P (𝐷 +P 𝑆))) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
20 addasspr 10046 . . . . 5 ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = (𝐴 +P (𝐹 +P (𝐷 +P 𝑆)))
21 addasspr 10046 . . . . 5 ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆)) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
2219, 20, 213eqtr4i 2803 . . . 4 ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆))
23 addcompr 10045 . . . . . . . 8 (𝐺 +P 𝐶) = (𝐶 +P 𝐺)
2423oveq1i 6803 . . . . . . 7 ((𝐺 +P 𝐶) +P 𝑅) = ((𝐶 +P 𝐺) +P 𝑅)
25 addasspr 10046 . . . . . . 7 ((𝐺 +P 𝐶) +P 𝑅) = (𝐺 +P (𝐶 +P 𝑅))
26 addasspr 10046 . . . . . . 7 ((𝐶 +P 𝐺) +P 𝑅) = (𝐶 +P (𝐺 +P 𝑅))
2724, 25, 263eqtr3i 2801 . . . . . 6 (𝐺 +P (𝐶 +P 𝑅)) = (𝐶 +P (𝐺 +P 𝑅))
2827oveq2i 6804 . . . . 5 (𝐵 +P (𝐺 +P (𝐶 +P 𝑅))) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
29 addasspr 10046 . . . . 5 ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = (𝐵 +P (𝐺 +P (𝐶 +P 𝑅)))
30 addasspr 10046 . . . . 5 ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅)) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
3128, 29, 303eqtr4i 2803 . . . 4 ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅))
3222, 31eqeq12i 2785 . . 3 (((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) ↔ ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆)) = ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅)))
3313, 32syl6bb 276 . 2 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩ ↔ ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆)) = ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅))))
341, 33syl5ibr 236 1 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cop 4322   class class class wbr 4786  (class class class)co 6793  Pcnp 9883   +P cpp 9885   ~R cer 9888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ni 9896  df-pli 9897  df-mi 9898  df-lti 9899  df-plpq 9932  df-mpq 9933  df-ltpq 9934  df-enq 9935  df-nq 9936  df-erq 9937  df-plq 9938  df-mq 9939  df-1nq 9940  df-rq 9941  df-ltnq 9942  df-np 10005  df-plp 10007  df-enr 10079
This theorem is referenced by:  addsrmo  10096
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