Theorem addcmpblnr 10981

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 7365	. 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 10930	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐹 ∈ P) → (𝐴 +P 𝐹) ∈ P)
3		addclpr 10930	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐺 ∈ P) → (𝐵 +P 𝐺) ∈ P)
4	2, 3	anim12i 614	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐹 ∈ P) ∧ (𝐵 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
5	4	an4s 661	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐹 ∈ P ∧ 𝐺 ∈ P)) → ((𝐴 +P 𝐹) ∈ P ∧ (𝐵 +P 𝐺) ∈ P))
6		addclpr 10930	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝑅 ∈ P) → (𝐶 +P 𝑅) ∈ P)
7		addclpr 10930	. . . . . . . 8 ⊢ ((𝐷 ∈ P ∧ 𝑆 ∈ P) → (𝐷 +P 𝑆) ∈ P)
8	6, 7	anim12i 614	. . . . . . 7 ⊢ (((𝐶 ∈ P ∧ 𝑅 ∈ P) ∧ (𝐷 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
9	8	an4s 661	. . . . . 6 ⊢ (((𝐶 ∈ P ∧ 𝐷 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P)) → ((𝐶 +P 𝑅) ∈ P ∧ (𝐷 +P 𝑆) ∈ P))
10	5, 9	anim12i 614	. . . . 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 661	. . . 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 10977	. . . 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 10933	. . . . . . . 8 ⊢ (𝐹 +P 𝐷) = (𝐷 +P 𝐹)
15	14	oveq1i 7366	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = ((𝐷 +P 𝐹) +P 𝑆)
16		addasspr 10934	. . . . . . 7 ⊢ ((𝐹 +P 𝐷) +P 𝑆) = (𝐹 +P (𝐷 +P 𝑆))
17		addasspr 10934	. . . . . . 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 7367	. . . . 5 ⊢ (𝐴 +P (𝐹 +P (𝐷 +P 𝑆))) = (𝐴 +P (𝐷 +P (𝐹 +P 𝑆)))
20		addasspr 10934	. . . . 5 ⊢ ((𝐴 +P 𝐹) +P (𝐷 +P 𝑆)) = (𝐴 +P (𝐹 +P (𝐷 +P 𝑆)))
21		addasspr 10934	. . . . 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 10933	. . . . . . . 8 ⊢ (𝐺 +P 𝐶) = (𝐶 +P 𝐺)
24	23	oveq1i 7366	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = ((𝐶 +P 𝐺) +P 𝑅)
25		addasspr 10934	. . . . . . 7 ⊢ ((𝐺 +P 𝐶) +P 𝑅) = (𝐺 +P (𝐶 +P 𝑅))
26		addasspr 10934	. . . . . . 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 7367	. . . . 5 ⊢ (𝐵 +P (𝐺 +P (𝐶 +P 𝑅))) = (𝐵 +P (𝐶 +P (𝐺 +P 𝑅)))
29		addasspr 10934	. . . . 5 ⊢ ((𝐵 +P 𝐺) +P (𝐶 +P 𝑅)) = (𝐵 +P (𝐺 +P (𝐶 +P 𝑅)))
30		addasspr 10934	. . . . 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 1542   ∈ wcel 2114  ⟨cop 4563   class class class wbr 5074  (class class class)co 7356  Pcnp 10771   +_P cpp 10773   ~_R cer 10776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-inf2 9551

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-oadd 8398  df-omul 8399  df-er 8632  df-ni 10784  df-pli 10785  df-mi 10786  df-lti 10787  df-plpq 10820  df-mpq 10821  df-ltpq 10822  df-enq 10823  df-nq 10824  df-erq 10825  df-plq 10826  df-mq 10827  df-1nq 10828  df-rq 10829  df-ltnq 10830  df-np 10893  df-plp 10895  df-enr 10967

This theorem is referenced by:  addsrmo  10985