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Theorem addcanprleml 6445
Description: Lemma for addcanprg 6447. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprleml (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) ⊆ (1st𝐶))

Proof of Theorem addcanprleml
Dummy variables f g 𝑟 𝑠 𝑡 u v w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6323 . . . . . . 7 (B P → ⟨(1stB), (2ndB)⟩ P)
2 prnmaddl 6338 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P v (1stB)) → w Q (v +Q w) (1stB))
31, 2sylan 267 . . . . . 6 ((B P v (1stB)) → w Q (v +Q w) (1stB))
433ad2antl2 1053 . . . . 5 (((A P B P 𝐶 P) v (1stB)) → w Q (v +Q w) (1stB))
54adantlr 449 . . . 4 ((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) → w Q (v +Q w) (1stB))
6 simprl 471 . . . . . 6 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) → w Q)
7 halfnqq 6261 . . . . . 6 (w Q𝑡 Q (𝑡 +Q 𝑡) = w)
86, 7syl 14 . . . . 5 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) → 𝑡 Q (𝑡 +Q 𝑡) = w)
9 simplll 473 . . . . . . . . . 10 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) → (A P B P 𝐶 P))
109adantr 261 . . . . . . . . 9 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → (A P B P 𝐶 P))
1110simp1d 902 . . . . . . . 8 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → A P)
12 prop 6323 . . . . . . . 8 (A P → ⟨(1stA), (2ndA)⟩ P)
1311, 12syl 14 . . . . . . 7 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → ⟨(1stA), (2ndA)⟩ P)
14 simprl 471 . . . . . . 7 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → 𝑡 Q)
15 prarloc2 6352 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P 𝑡 Q) → u (1stA)(u +Q 𝑡) (2ndA))
1613, 14, 15syl2anc 393 . . . . . 6 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → u (1stA)(u +Q 𝑡) (2ndA))
179ad2antrr 460 . . . . . . . . . . . 12 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (A P B P 𝐶 P))
1817simp1d 902 . . . . . . . . . . 11 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → A P)
1917simp2d 903 . . . . . . . . . . 11 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → B P)
20 addclpr 6386 . . . . . . . . . . 11 ((A P B P) → (A +P B) P)
2118, 19, 20syl2anc 393 . . . . . . . . . 10 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (A +P B) P)
22 prop 6323 . . . . . . . . . 10 ((A +P B) P → ⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P)
2321, 22syl 14 . . . . . . . . 9 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P)
2418, 12syl 14 . . . . . . . . . . 11 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1stA), (2ndA)⟩ P)
25 simprl 471 . . . . . . . . . . 11 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → u (1stA))
26 elprnql 6329 . . . . . . . . . . 11 ((⟨(1stA), (2ndA)⟩ P u (1stA)) → u Q)
2724, 25, 26syl2anc 393 . . . . . . . . . 10 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → u Q)
2819, 1syl 14 . . . . . . . . . . . 12 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1stB), (2ndB)⟩ P)
29 simplr 470 . . . . . . . . . . . . 13 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) → v (1stB))
3029ad2antrr 460 . . . . . . . . . . . 12 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → v (1stB))
31 elprnql 6329 . . . . . . . . . . . 12 ((⟨(1stB), (2ndB)⟩ P v (1stB)) → v Q)
3228, 30, 31syl2anc 393 . . . . . . . . . . 11 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → v Q)
33 simplrl 475 . . . . . . . . . . . 12 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → w Q)
3433adantr 261 . . . . . . . . . . 11 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → w Q)
35 addclnq 6228 . . . . . . . . . . 11 ((v Q w Q) → (v +Q w) Q)
3632, 34, 35syl2anc 393 . . . . . . . . . 10 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (v +Q w) Q)
37 addclnq 6228 . . . . . . . . . 10 ((u Q (v +Q w) Q) → (u +Q (v +Q w)) Q)
3827, 36, 37syl2anc 393 . . . . . . . . 9 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (u +Q (v +Q w)) Q)
39 prdisj 6340 . . . . . . . . 9 ((⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P (u +Q (v +Q w)) Q) → ¬ ((u +Q (v +Q w)) (1st ‘(A +P B)) (u +Q (v +Q w)) (2nd ‘(A +P B))))
4023, 38, 39syl2anc 393 . . . . . . . 8 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ¬ ((u +Q (v +Q w)) (1st ‘(A +P B)) (u +Q (v +Q w)) (2nd ‘(A +P B))))
4118adantr 261 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → A P)
4219adantr 261 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → B P)
43 simplrl 475 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → u (1stA))
44 simplrr 476 . . . . . . . . . . 11 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → (v +Q w) (1stB))
4544ad2antrr 460 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (v +Q w) (1stB))
46 df-iplp 6316 . . . . . . . . . . . 12 +P = (𝑟 P, 𝑠 P ↦ ⟨{f Qg Q Q (g (1st𝑟) (1st𝑠) f = (g +Q ))}, {f Qg Q Q (g (2nd𝑟) (2nd𝑠) f = (g +Q ))}⟩)
47 addclnq 6228 . . . . . . . . . . . 12 ((g Q Q) → (g +Q ) Q)
4846, 47genpprecll 6362 . . . . . . . . . . 11 ((A P B P) → ((u (1stA) (v +Q w) (1stB)) → (u +Q (v +Q w)) (1st ‘(A +P B))))
4948imp 115 . . . . . . . . . 10 (((A P B P) (u (1stA) (v +Q w) (1stB))) → (u +Q (v +Q w)) (1st ‘(A +P B)))
5041, 42, 43, 45, 49syl22anc 1120 . . . . . . . . 9 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (u +Q (v +Q w)) (1st ‘(A +P B)))
5127adantr 261 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → u Q)
5214ad2antrr 460 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → 𝑡 Q)
5332adantr 261 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → v Q)
54 addcomnqg 6234 . . . . . . . . . . . . . 14 ((f Q g Q) → (f +Q g) = (g +Q f))
5554adantl 262 . . . . . . . . . . . . 13 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) (f Q g Q)) → (f +Q g) = (g +Q f))
56 addassnqg 6235 . . . . . . . . . . . . . 14 ((f Q g Q Q) → ((f +Q g) +Q ) = (f +Q (g +Q )))
5756adantl 262 . . . . . . . . . . . . 13 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) (f Q g Q Q)) → ((f +Q g) +Q ) = (f +Q (g +Q )))
58 addclnq 6228 . . . . . . . . . . . . . 14 ((f Q g Q) → (f +Q g) Q)
5958adantl 262 . . . . . . . . . . . . 13 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) (f Q g Q)) → (f +Q g) Q)
6051, 52, 53, 55, 57, 52, 59caov4d 5604 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → ((u +Q 𝑡) +Q (v +Q 𝑡)) = ((u +Q v) +Q (𝑡 +Q 𝑡)))
61 simprr 472 . . . . . . . . . . . . . 14 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → (𝑡 +Q 𝑡) = w)
6261ad2antrr 460 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (𝑡 +Q 𝑡) = w)
6362oveq2d 5448 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → ((u +Q v) +Q (𝑡 +Q 𝑡)) = ((u +Q v) +Q w))
6433ad2antrr 460 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → w Q)
65 addassnqg 6235 . . . . . . . . . . . . 13 ((u Q v Q w Q) → ((u +Q v) +Q w) = (u +Q (v +Q w)))
6651, 53, 64, 65syl3anc 1119 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → ((u +Q v) +Q w) = (u +Q (v +Q w)))
6760, 63, 663eqtrd 2054 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → ((u +Q 𝑡) +Q (v +Q 𝑡)) = (u +Q (v +Q w)))
68 simplrr 476 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (u +Q 𝑡) (2ndA))
69 simpr 103 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (v +Q 𝑡) (2nd𝐶))
7017simp3d 904 . . . . . . . . . . . . . 14 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝐶 P)
7170adantr 261 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → 𝐶 P)
7246, 47genppreclu 6363 . . . . . . . . . . . . 13 ((A P 𝐶 P) → (((u +Q 𝑡) (2ndA) (v +Q 𝑡) (2nd𝐶)) → ((u +Q 𝑡) +Q (v +Q 𝑡)) (2nd ‘(A +P 𝐶))))
7341, 71, 72syl2anc 393 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (((u +Q 𝑡) (2ndA) (v +Q 𝑡) (2nd𝐶)) → ((u +Q 𝑡) +Q (v +Q 𝑡)) (2nd ‘(A +P 𝐶))))
7468, 69, 73mp2and 411 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → ((u +Q 𝑡) +Q (v +Q 𝑡)) (2nd ‘(A +P 𝐶)))
7567, 74eqeltrrd 2093 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (u +Q (v +Q w)) (2nd ‘(A +P 𝐶)))
76 simpr 103 . . . . . . . . . . . . 13 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (A +P B) = (A +P 𝐶))
7776ad3antrrr 461 . . . . . . . . . . . 12 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → (A +P B) = (A +P 𝐶))
7877ad2antrr 460 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (A +P B) = (A +P 𝐶))
79 fveq2 5099 . . . . . . . . . . . 12 ((A +P B) = (A +P 𝐶) → (2nd ‘(A +P B)) = (2nd ‘(A +P 𝐶)))
8079eleq2d 2085 . . . . . . . . . . 11 ((A +P B) = (A +P 𝐶) → ((u +Q (v +Q w)) (2nd ‘(A +P B)) ↔ (u +Q (v +Q w)) (2nd ‘(A +P 𝐶))))
8178, 80syl 14 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → ((u +Q (v +Q w)) (2nd ‘(A +P B)) ↔ (u +Q (v +Q w)) (2nd ‘(A +P 𝐶))))
8275, 81mpbird 156 . . . . . . . . 9 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → (u +Q (v +Q w)) (2nd ‘(A +P B)))
8350, 82jca 290 . . . . . . . 8 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (v +Q 𝑡) (2nd𝐶)) → ((u +Q (v +Q w)) (1st ‘(A +P B)) (u +Q (v +Q w)) (2nd ‘(A +P B))))
8440, 83mtand 578 . . . . . . 7 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ¬ (v +Q 𝑡) (2nd𝐶))
85 prop 6323 . . . . . . . . 9 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
8670, 85syl 14 . . . . . . . 8 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1st𝐶), (2nd𝐶)⟩ P)
87 simplrl 475 . . . . . . . . 9 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝑡 Q)
88 ltaddnq 6259 . . . . . . . . 9 ((v Q 𝑡 Q) → v <Q (v +Q 𝑡))
8932, 87, 88syl2anc 393 . . . . . . . 8 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → v <Q (v +Q 𝑡))
90 prloc 6339 . . . . . . . 8 ((⟨(1st𝐶), (2nd𝐶)⟩ P v <Q (v +Q 𝑡)) → (v (1st𝐶) (v +Q 𝑡) (2nd𝐶)))
9186, 89, 90syl2anc 393 . . . . . . 7 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (v (1st𝐶) (v +Q 𝑡) (2nd𝐶)))
9284, 91ecased 1222 . . . . . 6 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → v (1st𝐶))
9316, 92rexlimddv 2411 . . . . 5 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) (𝑡 Q (𝑡 +Q 𝑡) = w)) → v (1st𝐶))
948, 93rexlimddv 2411 . . . 4 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) → v (1st𝐶))
955, 94rexlimddv 2411 . . 3 ((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) → v (1st𝐶))
9695ex 108 . 2 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (v (1stB) → v (1st𝐶)))
9796ssrdv 2924 1 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) ⊆ (1st𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616   w3a 871   = wceq 1226   wcel 1370  wrex 2281  wss 2890  cop 3349   class class class wbr 3734  cfv 4825  (class class class)co 5432  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   +Q cplq 6136   <Q cltq 6139  Pcnp 6145   +P cpp 6147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-eprel 3996  df-id 4000  df-po 4003  df-iso 4004  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-1o 5912  df-2o 5913  df-oadd 5916  df-omul 5917  df-er 6013  df-ec 6015  df-qs 6019  df-ni 6158  df-pli 6159  df-mi 6160  df-lti 6161  df-plpq 6197  df-mpq 6198  df-enq 6200  df-nqqs 6201  df-plqqs 6202  df-mqqs 6203  df-1nqqs 6204  df-rq 6205  df-ltnqqs 6206  df-enq0 6273  df-nq0 6274  df-0nq0 6275  df-plq0 6276  df-mq0 6277  df-inp 6314  df-iplp 6316
This theorem is referenced by:  addcanprg  6447
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