Theorem addcmpblnr 11083

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 7414	. 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 11032	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐹 ∈ P) → (𝐴 +P 𝐹) ∈ P)
3		addclpr 11032	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐺 ∈ P) → (𝐵 +P 𝐺) ∈ P)
4	2, 3	anim12i 613	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐹 ∈ P) ∧ (𝐵 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
5	4	an4s 660	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐹 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
6		addclpr 11032	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝑅 ∈ P) → (𝐶 +P 𝑅) ∈ P)
7		addclpr 11032	. . . . . . . 8 ⊢ ((𝐷 ∈ P ∧ 𝑆 ∈ P) → (𝐷 +P 𝑆) ∈ P)
8	6, 7	anim12i 613	. . . . . . 7 ⊢ (((𝐶 ∈ P ∧ 𝑅 ∈ P) ∧ (𝐷 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
9	8	an4s 660	. . . . . 6 ⊢ (((𝐶 ∈ P ∧ 𝐷 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
10	5, 9	anim12i 613	. . . . 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 660	. . . 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 11079	. . . 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 11035	. . . . . . . 8 ⊢ (𝐹 +P 𝐷) = (𝐷 +P 𝐹)
15	14	oveq1i 7415	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = ((𝐷 +P 𝐹) +P 𝑆)
16		addasspr 11036	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = (𝐹 +P (𝐷 +P 𝑆))
17		addasspr 11036	. . . . . . 7 ⊢ ((𝐷 +P 𝐹) +P 𝑆) = (𝐷 +P (𝐹 +P 𝑆))
18	15, 16, 17	3eqtr3i 2766	. . . . . 6 ⊢ (𝐹 +P (𝐷 +P 𝑆)) = (𝐷 +P (𝐹 +P 𝑆))
19	18	oveq2i 7416	. . . . 5 ⊢ (𝐴 +P (𝐹 +P (𝐷 +P 𝑆))) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
20		addasspr 11036	. . . . 5 ⊢ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = (𝐴 +P (𝐹 +P (𝐷 +P 𝑆)))
21		addasspr 11036	. . . . 5 ⊢ ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆)) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
22	19, 20, 21	3eqtr4i 2768	. . . 4 ⊢ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆))
23		addcompr 11035	. . . . . . . 8 ⊢ (𝐺 +P 𝐶) = (𝐶 +P 𝐺)
24	23	oveq1i 7415	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = ((𝐶 +P 𝐺) +P 𝑅)
25		addasspr 11036	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = (𝐺 +P (𝐶 +P 𝑅))
26		addasspr 11036	. . . . . . 7 ⊢ ((𝐶 +P 𝐺) +P 𝑅) = (𝐶 +P (𝐺 +P 𝑅))
27	24, 25, 26	3eqtr3i 2766	. . . . . 6 ⊢ (𝐺 +P (𝐶 +P 𝑅)) = (𝐶 +P (𝐺 +P 𝑅))
28	27	oveq2i 7416	. . . . 5 ⊢ (𝐵 +P (𝐺 +P (𝐶 +P 𝑅))) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
29		addasspr 11036	. . . . 5 ⊢ ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = (𝐵 +P (𝐺 +P (𝐶 +P 𝑅)))
30		addasspr 11036	. . . . 5 ⊢ ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅)) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
31	28, 29, 30	3eqtr4i 2768	. . . 4 ⊢ ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅))
32	22, 31	eqeq12i 2753	. . 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	bitrdi 287	. 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	imbitrrid 246	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⟨cop 4607   class class class wbr 5119  (class class class)co 7405  Pcnp 10873   +_P cpp 10875   ~_R cer 10878

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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8719  df-ni 10886  df-pli 10887  df-mi 10888  df-lti 10889  df-plpq 10922  df-mpq 10923  df-ltpq 10924  df-enq 10925  df-nq 10926  df-erq 10927  df-plq 10928  df-mq 10929  df-1nq 10930  df-rq 10931  df-ltnq 10932  df-np 10995  df-plp 10997  df-enr 11069

This theorem is referenced by:  addsrmo  11087