Theorem addcmpblnr 10287

Description: Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addcmpblnr	⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((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

Step	Hyp	Ref	Expression
1		oveq12 6983	. 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 10236	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐹 ∈ P) → (𝐴 +P 𝐹) ∈ P)
3		addclpr 10236	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐺 ∈ P) → (𝐵 +P 𝐺) ∈ P)
4	2, 3	anim12i 604	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐹 ∈ P) ∧ (𝐵 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
5	4	an4s 648	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐹 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
6		addclpr 10236	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝑅 ∈ P) → (𝐶 +P 𝑅) ∈ P)
7		addclpr 10236	. . . . . . . 8 ⊢ ((𝐷 ∈ P ∧ 𝑆 ∈ P) → (𝐷 +P 𝑆) ∈ P)
8	6, 7	anim12i 604	. . . . . . 7 ⊢ (((𝐶 ∈ P ∧ 𝑅 ∈ P) ∧ (𝐷 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
9	8	an4s 648	. . . . . 6 ⊢ (((𝐶 ∈ P ∧ 𝐷 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
10	5, 9	anim12i 604	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐹 ∈ P ∧ 𝐺 ∈ P)) ∧ ((𝐶 ∈ P ∧ 𝐷 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P) ∧ ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P)))
11	10	an4s 648	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P) ∧ ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P)))
12		enrbreq 10283	. . . 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))))
13	11, 12	syl 17	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ 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))))
14		addcompr 10239	. . . . . . . 8 ⊢ (𝐹 +P 𝐷) = (𝐷 +P 𝐹)
15	14	oveq1i 6984	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = ((𝐷 +P 𝐹) +P 𝑆)
16		addasspr 10240	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = (𝐹 +P (𝐷 +P 𝑆))
17		addasspr 10240	. . . . . . 7 ⊢ ((𝐷 +P 𝐹) +P 𝑆) = (𝐷 +P (𝐹 +P 𝑆))
18	15, 16, 17	3eqtr3i 2803	. . . . . 6 ⊢ (𝐹 +P (𝐷 +P 𝑆)) = (𝐷 +P (𝐹 +P 𝑆))
19	18	oveq2i 6985	. . . . 5 ⊢ (𝐴 +P (𝐹 +P (𝐷 +P 𝑆))) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
20		addasspr 10240	. . . . 5 ⊢ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = (𝐴 +P (𝐹 +P (𝐷 +P 𝑆)))
21		addasspr 10240	. . . . 5 ⊢ ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆)) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
22	19, 20, 21	3eqtr4i 2805	. . . 4 ⊢ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆))
23		addcompr 10239	. . . . . . . 8 ⊢ (𝐺 +P 𝐶) = (𝐶 +P 𝐺)
24	23	oveq1i 6984	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = ((𝐶 +P 𝐺) +P 𝑅)
25		addasspr 10240	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = (𝐺 +P (𝐶 +P 𝑅))
26		addasspr 10240	. . . . . . 7 ⊢ ((𝐶 +P 𝐺) +P 𝑅) = (𝐶 +P (𝐺 +P 𝑅))
27	24, 25, 26	3eqtr3i 2803	. . . . . 6 ⊢ (𝐺 +P (𝐶 +P 𝑅)) = (𝐶 +P (𝐺 +P 𝑅))
28	27	oveq2i 6985	. . . . 5 ⊢ (𝐵 +P (𝐺 +P (𝐶 +P 𝑅))) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
29		addasspr 10240	. . . . 5 ⊢ ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = (𝐵 +P (𝐺 +P (𝐶 +P 𝑅)))
30		addasspr 10240	. . . . 5 ⊢ ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅)) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
31	28, 29, 30	3eqtr4i 2805	. . . 4 ⊢ ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅))
32	22, 31	eqeq12i 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)))
33	13, 32	syl6bb 279	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (⟨(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))))
34	1, 33	syl5ibr 238	1 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((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)⟩))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ⟨cop 4441   class class class wbr 4925  (class class class)co 6974  Pcnp 10077   +_P cpp 10079   ~_R cer 10082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-inf2 8896

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-omul 7908  df-er 8087  df-ni 10090  df-pli 10091  df-mi 10092  df-lti 10093  df-plpq 10126  df-mpq 10127  df-ltpq 10128  df-enq 10129  df-nq 10130  df-erq 10131  df-plq 10132  df-mq 10133  df-1nq 10134  df-rq 10135  df-ltnq 10136  df-np 10199  df-plp 10201  df-enr 10273

This theorem is referenced by:  addsrmo  10291