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Theorem addcanprleml 6586
Description: Lemma for addcanprg 6588. (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 6457 . . . . . . 7 (B P → ⟨(1stB), (2ndB)⟩ P)
2 prnmaddl 6472 . . . . . . 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 1066 . . . . 5 (((A P B P 𝐶 P) v (1stB)) → w Q (v +Q w) (1stB))
54adantlr 446 . . . 4 ((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) → w Q (v +Q w) (1stB))
6 simprl 483 . . . . . 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 6393 . . . . . 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 485 . . . . . . . . . 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 915 . . . . . . . 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 6457 . . . . . . . 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 483 . . . . . . 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 6486 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P 𝑡 Q) → u (1stA)(u +Q 𝑡) (2ndA))
1613, 14, 15syl2anc 391 . . . . . 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 457 . . . . . . . . . . . 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 915 . . . . . . . . . . 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 916 . . . . . . . . . . 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 6520 . . . . . . . . . . 11 ((A P B P) → (A +P B) P)
2118, 19, 20syl2anc 391 . . . . . . . . . 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 6457 . . . . . . . . . 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 483 . . . . . . . . . . 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 6463 . . . . . . . . . . 11 ((⟨(1stA), (2ndA)⟩ P u (1stA)) → u Q)
2724, 25, 26syl2anc 391 . . . . . . . . . 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 482 . . . . . . . . . . . . 13 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (1stB)) (w Q (v +Q w) (1stB))) → v (1stB))
3029ad2antrr 457 . . . . . . . . . . . 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 6463 . . . . . . . . . . . 12 ((⟨(1stB), (2ndB)⟩ P v (1stB)) → v Q)
3228, 30, 31syl2anc 391 . . . . . . . . . . 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 487 . . . . . . . . . . . 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 6359 . . . . . . . . . . 11 ((v Q w Q) → (v +Q w) Q)
3632, 34, 35syl2anc 391 . . . . . . . . . 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 6359 . . . . . . . . . 10 ((u Q (v +Q w) Q) → (u +Q (v +Q w)) Q)
3827, 36, 37syl2anc 391 . . . . . . . . 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 6474 . . . . . . . . 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 391 . . . . . . . 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 487 . . . . . . . . . 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 488 . . . . . . . . . . 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 457 . . . . . . . . . 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 6450 . . . . . . . . . . . 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 6359 . . . . . . . . . . . 12 ((g Q Q) → (g +Q ) Q)
4846, 47genpprecll 6496 . . . . . . . . . . 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 1135 . . . . . . . . 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 457 . . . . . . . . . . . . 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 6365 . . . . . . . . . . . . . 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 6366 . . . . . . . . . . . . . 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 6359 . . . . . . . . . . . . . 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 5627 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 5471 . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 6366 . . . . . . . . . . . . 13 ((u Q v Q w Q) → ((u +Q v) +Q w) = (u +Q (v +Q w)))
6651, 53, 64, 65syl3anc 1134 . . . . . . . . . . . 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 2073 . . . . . . . . . . 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 488 . . . . . . . . . . . 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 917 . . . . . . . . . . . . . 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 6497 . . . . . . . . . . . . 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 391 . . . . . . . . . . . 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 409 . . . . . . . . . . 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 2112 . . . . . . . . . 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 457 . . . . . . . . . . 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 5121 . . . . . . . . . . . 12 ((A +P B) = (A +P 𝐶) → (2nd ‘(A +P B)) = (2nd ‘(A +P 𝐶)))
8079eleq2d 2104 . . . . . . . . . . 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 590 . . . . . . 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 6457 . . . . . . . . 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 487 . . . . . . . . 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 6390 . . . . . . . . 9 ((v Q 𝑡 Q) → v <Q (v +Q 𝑡))
8932, 87, 88syl2anc 391 . . . . . . . 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 6473 . . . . . . . 8 ((⟨(1st𝐶), (2nd𝐶)⟩ P v <Q (v +Q 𝑡)) → (v (1st𝐶) (v +Q 𝑡) (2nd𝐶)))
9186, 89, 90syl2anc 391 . . . . . . 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 1238 . . . . . 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 2431 . . . . 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 2431 . . . 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 2431 . . 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 2945 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 628   w3a 884   = wceq 1242   wcel 1390  wrex 2301  wss 2911  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275   +P cpp 6277
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450
This theorem is referenced by:  addcanprg  6588
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