Theorem addcmpblnr 11155

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 7429	. 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 11104	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐹 ∈ P) → (𝐴 +P 𝐹) ∈ P)
3		addclpr 11104	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐺 ∈ P) → (𝐵 +P 𝐺) ∈ P)
4	2, 3	anim12i 625	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐹 ∈ P) ∧ (𝐵 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
5	4	an4s 673	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐹 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
6		addclpr 11104	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝑅 ∈ P) → (𝐶 +P 𝑅) ∈ P)
7		addclpr 11104	. . . . . . . 8 ⊢ ((𝐷 ∈ P ∧ 𝑆 ∈ P) → (𝐷 +P 𝑆) ∈ P)
8	6, 7	anim12i 625	. . . . . . 7 ⊢ (((𝐶 ∈ P ∧ 𝑅 ∈ P) ∧ (𝐷 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
9	8	an4s 673	. . . . . 6 ⊢ (((𝐶 ∈ P ∧ 𝐷 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
10	5, 9	anim12i 625	. . . . 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 673	. . . 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 11151	. . . 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 18	. . 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 11107	. . . . . . . 8 ⊢ (𝐹 +P 𝐷) = (𝐷 +P 𝐹)
15	14	oveq1i 7430	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = ((𝐷 +P 𝐹) +P 𝑆)
16		addasspr 11108	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = (𝐹 +P (𝐷 +P 𝑆))
17		addasspr 11108	. . . . . . 7 ⊢ ((𝐷 +P 𝐹) +P 𝑆) = (𝐷 +P (𝐹 +P 𝑆))
18	15, 16, 17	3eqtr3i 2791	. . . . . 6 ⊢ (𝐹 +P (𝐷 +P 𝑆)) = (𝐷 +P (𝐹 +P 𝑆))
19	18	oveq2i 7431	. . . . 5 ⊢ (𝐴 +P (𝐹 +P (𝐷 +P 𝑆))) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
20		addasspr 11108	. . . . 5 ⊢ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = (𝐴 +P (𝐹 +P (𝐷 +P 𝑆)))
21		addasspr 11108	. . . . 5 ⊢ ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆)) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
22	19, 20, 21	3eqtr4i 2793	. . . 4 ⊢ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = ((𝐴 +P 𝐷) +P (𝐹 +P 𝑆))
23		addcompr 11107	. . . . . . . 8 ⊢ (𝐺 +P 𝐶) = (𝐶 +P 𝐺)
24	23	oveq1i 7430	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = ((𝐶 +P 𝐺) +P 𝑅)
25		addasspr 11108	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = (𝐺 +P (𝐶 +P 𝑅))
26		addasspr 11108	. . . . . . 7 ⊢ ((𝐶 +P 𝐺) +P 𝑅) = (𝐶 +P (𝐺 +P 𝑅))
27	24, 25, 26	3eqtr3i 2791	. . . . . 6 ⊢ (𝐺 +P (𝐶 +P 𝑅)) = (𝐶 +P (𝐺 +P 𝑅))
28	27	oveq2i 7431	. . . . 5 ⊢ (𝐵 +P (𝐺 +P (𝐶 +P 𝑅))) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
29		addasspr 11108	. . . . 5 ⊢ ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = (𝐵 +P (𝐺 +P (𝐶 +P 𝑅)))
30		addasspr 11108	. . . . 5 ⊢ ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅)) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
31	28, 29, 30	3eqtr4i 2793	. . . 4 ⊢ ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = ((𝐵 +P 𝐶) +P (𝐺 +P 𝑅))
32	22, 31	eqeq12i 2778	. . 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 290	. 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 249	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)⟩))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ⟨cop 4589   class class class wbr 5102  (class class class)co 7420  Pcnp 10945   +_P cpp 10947   ~_R cer 10950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-oadd 8481  df-omul 8482  df-er 8718  df-ni 10958  df-pli 10959  df-mi 10960  df-lti 10961  df-plpq 10994  df-mpq 10995  df-ltpq 10996  df-enq 10997  df-nq 10998  df-erq 10999  df-plq 11000  df-mq 11001  df-1nq 11002  df-rq 11003  df-ltnq 11004  df-np 11067  df-plp 11069  df-enr 11141

This theorem is used by:  addsrmo  11159