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Theorem mulcmpblnrlemg 7189
Description: Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
Assertion
Ref Expression
mulcmpblnrlemg ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))))

Proof of Theorem mulcmpblnrlemg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpllr 501 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝐵P)
2 simprlr 505 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝐺P)
3 mulclpr 7034 . . . . . . . . 9 ((𝐵P𝐺P) → (𝐵 ·P 𝐺) ∈ P)
41, 2, 3syl2anc 403 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐵 ·P 𝐺) ∈ P)
5 simplrr 503 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝐷P)
6 simprrl 506 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝑅P)
7 mulclpr 7034 . . . . . . . . 9 ((𝐷P𝑅P) → (𝐷 ·P 𝑅) ∈ P)
85, 6, 7syl2anc 403 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P 𝑅) ∈ P)
9 addclpr 6999 . . . . . . . 8 (((𝐵 ·P 𝐺) ∈ P ∧ (𝐷 ·P 𝑅) ∈ P) → ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) ∈ P)
104, 8, 9syl2anc 403 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) ∈ P)
11 simplrl 502 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝐶P)
12 mulclpr 7034 . . . . . . . 8 ((𝐶P𝐺P) → (𝐶 ·P 𝐺) ∈ P)
1311, 2, 12syl2anc 403 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P 𝐺) ∈ P)
14 simprll 504 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝐹P)
15 mulclpr 7034 . . . . . . . . 9 ((𝐵P𝐹P) → (𝐵 ·P 𝐹) ∈ P)
161, 14, 15syl2anc 403 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐵 ·P 𝐹) ∈ P)
17 mulclpr 7034 . . . . . . . . 9 ((𝐶P𝑅P) → (𝐶 ·P 𝑅) ∈ P)
1811, 6, 17syl2anc 403 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P 𝑅) ∈ P)
19 addclpr 6999 . . . . . . . 8 (((𝐵 ·P 𝐹) ∈ P ∧ (𝐶 ·P 𝑅) ∈ P) → ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)) ∈ P)
2016, 18, 19syl2anc 403 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)) ∈ P)
21 addassprg 7041 . . . . . . 7 ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) ∈ P ∧ (𝐶 ·P 𝐺) ∈ P ∧ ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)) ∈ P) → ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
2210, 13, 20, 21syl3anc 1170 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
2322adantr 270 . . . . 5 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
24 oveq2 5599 . . . . . . . . . . 11 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐷 ·P (𝐹 +P 𝑆)) = (𝐷 ·P (𝐺 +P 𝑅)))
2524ad2antll 475 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (𝐷 ·P (𝐹 +P 𝑆)) = (𝐷 ·P (𝐺 +P 𝑅)))
26 simprrr 507 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝑆P)
27 distrprg 7050 . . . . . . . . . . . 12 ((𝐷P𝐹P𝑆P) → (𝐷 ·P (𝐹 +P 𝑆)) = ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)))
285, 14, 26, 27syl3anc 1170 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P (𝐹 +P 𝑆)) = ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)))
2928adantr 270 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (𝐷 ·P (𝐹 +P 𝑆)) = ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)))
30 distrprg 7050 . . . . . . . . . . . 12 ((𝐷P𝐺P𝑅P) → (𝐷 ·P (𝐺 +P 𝑅)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
315, 2, 6, 30syl3anc 1170 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P (𝐺 +P 𝑅)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
3231adantr 270 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (𝐷 ·P (𝐺 +P 𝑅)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
3325, 29, 323eqtr3d 2123 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
3433oveq2d 5607 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))))
35 simplll 500 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → 𝐴P)
36 mulclpr 7034 . . . . . . . . . . 11 ((𝐴P𝐺P) → (𝐴 ·P 𝐺) ∈ P)
3735, 2, 36syl2anc 403 . . . . . . . . . 10 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐴 ·P 𝐺) ∈ P)
38 mulclpr 7034 . . . . . . . . . . 11 ((𝐷P𝐺P) → (𝐷 ·P 𝐺) ∈ P)
395, 2, 38syl2anc 403 . . . . . . . . . 10 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P 𝐺) ∈ P)
40 addassprg 7041 . . . . . . . . . 10 (((𝐴 ·P 𝐺) ∈ P ∧ (𝐷 ·P 𝐺) ∈ P ∧ (𝐷 ·P 𝑅) ∈ P) → (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))))
4137, 39, 8, 40syl3anc 1170 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))))
4241adantr 270 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))))
43 oveq1 5598 . . . . . . . . . . 11 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐵 +P 𝐶) ·P 𝐺))
4443ad2antrl 474 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐵 +P 𝐶) ·P 𝐺))
45 distrprg 7050 . . . . . . . . . . . . 13 ((𝐺P𝐴P𝐷P) → (𝐺 ·P (𝐴 +P 𝐷)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷)))
462, 35, 5, 45syl3anc 1170 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐺 ·P (𝐴 +P 𝐷)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷)))
47 addclpr 6999 . . . . . . . . . . . . . 14 ((𝐴P𝐷P) → (𝐴 +P 𝐷) ∈ P)
4835, 5, 47syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐴 +P 𝐷) ∈ P)
49 mulcomprg 7042 . . . . . . . . . . . . 13 (((𝐴 +P 𝐷) ∈ P𝐺P) → ((𝐴 +P 𝐷) ·P 𝐺) = (𝐺 ·P (𝐴 +P 𝐷)))
5048, 2, 49syl2anc 403 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 +P 𝐷) ·P 𝐺) = (𝐺 ·P (𝐴 +P 𝐷)))
51 mulcomprg 7042 . . . . . . . . . . . . . 14 ((𝐴P𝐺P) → (𝐴 ·P 𝐺) = (𝐺 ·P 𝐴))
5235, 2, 51syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐴 ·P 𝐺) = (𝐺 ·P 𝐴))
53 mulcomprg 7042 . . . . . . . . . . . . . 14 ((𝐷P𝐺P) → (𝐷 ·P 𝐺) = (𝐺 ·P 𝐷))
545, 2, 53syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P 𝐺) = (𝐺 ·P 𝐷))
5552, 54oveq12d 5609 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷)))
5646, 50, 553eqtr4d 2125 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)))
5756adantr 270 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)))
58 distrprg 7050 . . . . . . . . . . . . 13 ((𝐺P𝐵P𝐶P) → (𝐺 ·P (𝐵 +P 𝐶)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶)))
592, 1, 11, 58syl3anc 1170 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐺 ·P (𝐵 +P 𝐶)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶)))
60 addclpr 6999 . . . . . . . . . . . . . 14 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
611, 11, 60syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐵 +P 𝐶) ∈ P)
62 mulcomprg 7042 . . . . . . . . . . . . 13 (((𝐵 +P 𝐶) ∈ P𝐺P) → ((𝐵 +P 𝐶) ·P 𝐺) = (𝐺 ·P (𝐵 +P 𝐶)))
6361, 2, 62syl2anc 403 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 +P 𝐶) ·P 𝐺) = (𝐺 ·P (𝐵 +P 𝐶)))
64 mulcomprg 7042 . . . . . . . . . . . . . 14 ((𝐵P𝐺P) → (𝐵 ·P 𝐺) = (𝐺 ·P 𝐵))
651, 2, 64syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐵 ·P 𝐺) = (𝐺 ·P 𝐵))
66 mulcomprg 7042 . . . . . . . . . . . . . 14 ((𝐶P𝐺P) → (𝐶 ·P 𝐺) = (𝐺 ·P 𝐶))
6711, 2, 66syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P 𝐺) = (𝐺 ·P 𝐶))
6865, 67oveq12d 5609 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶)))
6959, 63, 683eqtr4d 2125 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 +P 𝐶) ·P 𝐺) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)))
7069adantr 270 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐵 +P 𝐶) ·P 𝐺) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)))
7144, 57, 703eqtr3d 2123 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)))
7271oveq1d 5606 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
7334, 42, 723eqtr2d 2121 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
74 mulclpr 7034 . . . . . . . . . 10 ((𝐷P𝐹P) → (𝐷 ·P 𝐹) ∈ P)
755, 14, 74syl2anc 403 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P 𝐹) ∈ P)
76 mulclpr 7034 . . . . . . . . . 10 ((𝐷P𝑆P) → (𝐷 ·P 𝑆) ∈ P)
775, 26, 76syl2anc 403 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P 𝑆) ∈ P)
78 addcomprg 7040 . . . . . . . . . 10 ((𝑥P𝑦P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
7978adantl 271 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝑥P𝑦P)) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
80 addassprg 7041 . . . . . . . . . 10 ((𝑥P𝑦P𝑧P) → ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧)))
8180adantl 271 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝑥P𝑦P𝑧P)) → ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧)))
8237, 75, 77, 79, 81caov12d 5761 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = ((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))))
8382adantr 270 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = ((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))))
844, 13, 8, 79, 81caov32d 5760 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)))
8584adantr 270 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)))
8673, 83, 853eqtr3d 2123 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)))
8786oveq1d 5606 . . . . 5 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
88 oveq1 5598 . . . . . . . . . . . 12 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐵 +P 𝐶) ·P 𝐹))
8988adantl 271 . . . . . . . . . . 11 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐵 +P 𝐶) ·P 𝐹))
90 distrprg 7050 . . . . . . . . . . . . . 14 ((𝐹P𝐴P𝐷P) → (𝐹 ·P (𝐴 +P 𝐷)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷)))
9114, 35, 5, 90syl3anc 1170 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐹 ·P (𝐴 +P 𝐷)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷)))
92 mulcomprg 7042 . . . . . . . . . . . . . 14 (((𝐴 +P 𝐷) ∈ P𝐹P) → ((𝐴 +P 𝐷) ·P 𝐹) = (𝐹 ·P (𝐴 +P 𝐷)))
9348, 14, 92syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 +P 𝐷) ·P 𝐹) = (𝐹 ·P (𝐴 +P 𝐷)))
94 mulcomprg 7042 . . . . . . . . . . . . . . 15 ((𝐴P𝐹P) → (𝐴 ·P 𝐹) = (𝐹 ·P 𝐴))
9535, 14, 94syl2anc 403 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐴 ·P 𝐹) = (𝐹 ·P 𝐴))
96 mulcomprg 7042 . . . . . . . . . . . . . . 15 ((𝐷P𝐹P) → (𝐷 ·P 𝐹) = (𝐹 ·P 𝐷))
975, 14, 96syl2anc 403 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐷 ·P 𝐹) = (𝐹 ·P 𝐷))
9895, 97oveq12d 5609 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷)))
9991, 93, 983eqtr4d 2125 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)))
10099adantr 270 . . . . . . . . . . 11 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)))
101 distrprg 7050 . . . . . . . . . . . . . 14 ((𝐹P𝐵P𝐶P) → (𝐹 ·P (𝐵 +P 𝐶)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶)))
10214, 1, 11, 101syl3anc 1170 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐹 ·P (𝐵 +P 𝐶)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶)))
103 mulcomprg 7042 . . . . . . . . . . . . . 14 (((𝐵 +P 𝐶) ∈ P𝐹P) → ((𝐵 +P 𝐶) ·P 𝐹) = (𝐹 ·P (𝐵 +P 𝐶)))
10461, 14, 103syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 +P 𝐶) ·P 𝐹) = (𝐹 ·P (𝐵 +P 𝐶)))
105 mulcomprg 7042 . . . . . . . . . . . . . . 15 ((𝐵P𝐹P) → (𝐵 ·P 𝐹) = (𝐹 ·P 𝐵))
1061, 14, 105syl2anc 403 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐵 ·P 𝐹) = (𝐹 ·P 𝐵))
107 mulcomprg 7042 . . . . . . . . . . . . . . 15 ((𝐶P𝐹P) → (𝐶 ·P 𝐹) = (𝐹 ·P 𝐶))
10811, 14, 107syl2anc 403 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P 𝐹) = (𝐹 ·P 𝐶))
109106, 108oveq12d 5609 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶)))
110102, 104, 1093eqtr4d 2125 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 +P 𝐶) ·P 𝐹) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)))
111110adantr 270 . . . . . . . . . . 11 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)) → ((𝐵 +P 𝐶) ·P 𝐹) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)))
11289, 100, 1113eqtr3d 2123 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)) → ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)))
113112oveq1d 5606 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)))
114113adantrr 463 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)))
115 mulclpr 7034 . . . . . . . . . . . . 13 ((𝐶P𝐹P) → (𝐶 ·P 𝐹) ∈ P)
11611, 14, 115syl2anc 403 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P 𝐹) ∈ P)
117 mulclpr 7034 . . . . . . . . . . . . 13 ((𝐶P𝑆P) → (𝐶 ·P 𝑆) ∈ P)
11811, 26, 117syl2anc 403 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P 𝑆) ∈ P)
119 addassprg 7041 . . . . . . . . . . . 12 (((𝐵 ·P 𝐹) ∈ P ∧ (𝐶 ·P 𝐹) ∈ P ∧ (𝐶 ·P 𝑆) ∈ P) → (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))))
12016, 116, 118, 119syl3anc 1170 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))))
121120adantr 270 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))))
122 oveq2 5599 . . . . . . . . . . . . 13 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐶 ·P (𝐹 +P 𝑆)) = (𝐶 ·P (𝐺 +P 𝑅)))
123122adantl 271 . . . . . . . . . . . 12 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (𝐶 ·P (𝐹 +P 𝑆)) = (𝐶 ·P (𝐺 +P 𝑅)))
124 distrprg 7050 . . . . . . . . . . . . . 14 ((𝐶P𝐹P𝑆P) → (𝐶 ·P (𝐹 +P 𝑆)) = ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)))
12511, 14, 26, 124syl3anc 1170 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P (𝐹 +P 𝑆)) = ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)))
126125adantr 270 . . . . . . . . . . . 12 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (𝐶 ·P (𝐹 +P 𝑆)) = ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)))
127 distrprg 7050 . . . . . . . . . . . . . 14 ((𝐶P𝐺P𝑅P) → (𝐶 ·P (𝐺 +P 𝑅)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
12811, 2, 6, 127syl3anc 1170 . . . . . . . . . . . . 13 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐶 ·P (𝐺 +P 𝑅)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
129128adantr 270 . . . . . . . . . . . 12 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (𝐶 ·P (𝐺 +P 𝑅)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
130123, 126, 1293eqtr3d 2123 . . . . . . . . . . 11 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
131130oveq2d 5607 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
132121, 131eqtrd 2115 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
133132adantrl 462 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
134114, 133eqtrd 2115 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
135 mulclpr 7034 . . . . . . . . . 10 ((𝐴P𝐹P) → (𝐴 ·P 𝐹) ∈ P)
13635, 14, 135syl2anc 403 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (𝐴 ·P 𝐹) ∈ P)
137136, 75, 118, 79, 81caov32d 5760 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹)))
138137adantr 270 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹)))
13916, 13, 18, 79, 81caov12d 5761 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))) = ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
140139adantr 270 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))) = ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
141134, 138, 1403eqtr3d 2123 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹)) = ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
142141oveq2d 5607 . . . . 5 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
14323, 87, 1423eqtr4rd 2126 . . . 4 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
144 addclpr 6999 . . . . . . 7 (((𝐴 ·P 𝐹) ∈ P ∧ (𝐶 ·P 𝑆) ∈ P) → ((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) ∈ P)
145136, 118, 144syl2anc 403 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) ∈ P)
14610, 145, 75, 79, 81caov13d 5763 . . . . 5 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))))
147146adantr 270 . . . 4 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))))
148 addclpr 6999 . . . . . . 7 (((𝐴 ·P 𝐺) ∈ P ∧ (𝐷 ·P 𝑆) ∈ P) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) ∈ P)
14937, 77, 148syl2anc 403 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) ∈ P)
150 addassprg 7041 . . . . . 6 (((𝐷 ·P 𝐹) ∈ P ∧ ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) ∈ P ∧ ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)) ∈ P) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
15175, 149, 20, 150syl3anc 1170 . . . . 5 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
152151adantr 270 . . . 4 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
153143, 147, 1523eqtr3d 2123 . . 3 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
154 addclpr 6999 . . . . . . 7 ((𝑥P𝑦P) → (𝑥 +P 𝑦) ∈ P)
155154adantl 271 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ (𝑥P𝑦P)) → (𝑥 +P 𝑦) ∈ P)
156136, 118, 4, 79, 81, 8, 155caov4d 5764 . . . . 5 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅))) = (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))))
157156oveq2d 5607 . . . 4 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))))
158157adantr 270 . . 3 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))))
15937, 77, 16, 79, 81, 18, 155caov42d 5766 . . . . 5 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))
160159oveq2d 5607 . . . 4 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))
161160adantr 270 . . 3 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))
162153, 158, 1613eqtr3d 2123 . 2 (((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) ∧ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅))) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))
163162ex 113 1 ((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  (class class class)co 5591  Pcnp 6753   +P cpp 6755   ·P cmp 6756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-imp 6931
This theorem is referenced by:  mulcmpblnr  7190
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