ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addcanprlemu Structured version   GIF version

Theorem addcanprlemu 6587
Description: Lemma for addcanprg 6588. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprlemu (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2ndB) ⊆ (2nd𝐶))

Proof of Theorem addcanprlemu
Dummy variables f g 𝑞 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 prnminu 6471 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P v (2ndB)) → 𝑟 (2ndB)𝑟 <Q v)
31, 2sylan 267 . . . . . 6 ((B P v (2ndB)) → 𝑟 (2ndB)𝑟 <Q v)
433ad2antl2 1066 . . . . 5 (((A P B P 𝐶 P) v (2ndB)) → 𝑟 (2ndB)𝑟 <Q v)
54adantlr 446 . . . 4 ((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) → 𝑟 (2ndB)𝑟 <Q v)
6 simprr 484 . . . . . 6 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) → 𝑟 <Q v)
7 ltexnqi 6392 . . . . . 6 (𝑟 <Q vw Q (𝑟 +Q w) = v)
86, 7syl 14 . . . . 5 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) → w Q (𝑟 +Q w) = v)
9 simprl 483 . . . . . . 7 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) → w Q)
10 halfnqq 6393 . . . . . . 7 (w Q𝑡 Q (𝑡 +Q 𝑡) = w)
119, 10syl 14 . . . . . 6 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) → 𝑡 Q (𝑡 +Q 𝑡) = w)
12 prop 6457 . . . . . . . . . . . . . 14 (A P → ⟨(1stA), (2ndA)⟩ P)
13 prarloc2 6486 . . . . . . . . . . . . . 14 ((⟨(1stA), (2ndA)⟩ P 𝑡 Q) → u (1stA)(u +Q 𝑡) (2ndA))
1412, 13sylan 267 . . . . . . . . . . . . 13 ((A P 𝑡 Q) → u (1stA)(u +Q 𝑡) (2ndA))
1514adantrr 448 . . . . . . . . . . . 12 ((A P (𝑡 Q (𝑡 +Q 𝑡) = w)) → u (1stA)(u +Q 𝑡) (2ndA))
16153ad2antl1 1065 . . . . . . . . . . 11 (((A P B P 𝐶 P) (𝑡 Q (𝑡 +Q 𝑡) = w)) → u (1stA)(u +Q 𝑡) (2ndA))
1716adantlr 446 . . . . . . . . . 10 ((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) (𝑡 Q (𝑡 +Q 𝑡) = w)) → u (1stA)(u +Q 𝑡) (2ndA))
1817adantlr 446 . . . . . . . . 9 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑡 Q (𝑡 +Q 𝑡) = w)) → u (1stA)(u +Q 𝑡) (2ndA))
1918adantlr 446 . . . . . . . 8 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) → u (1stA)(u +Q 𝑡) (2ndA))
2019adantlr 446 . . . . . . 7 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) → u (1stA)(u +Q 𝑡) (2ndA))
21 simplll 485 . . . . . . . . . . . . . 14 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) → (A P B P 𝐶 P))
2221ad3antrrr 461 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (A P B P 𝐶 P))
2322simp1d 915 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → A P)
2422simp2d 916 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → B P)
25 addclpr 6520 . . . . . . . . . . . 12 ((A P B P) → (A +P B) P)
2623, 24, 25syl2anc 391 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (A +P B) P)
27 prop 6457 . . . . . . . . . . 11 ((A +P B) P → ⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P)
2826, 27syl 14 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P)
2923, 12syl 14 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1stA), (2ndA)⟩ P)
30 simprl 483 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → u (1stA))
31 elprnql 6463 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P u (1stA)) → u Q)
3229, 30, 31syl2anc 391 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → u Q)
33 simplrl 487 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝑡 Q)
34 addclnq 6359 . . . . . . . . . . . 12 ((u Q 𝑡 Q) → (u +Q 𝑡) Q)
3532, 33, 34syl2anc 391 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (u +Q 𝑡) Q)
3624, 1syl 14 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1stB), (2ndB)⟩ P)
37 simprl 483 . . . . . . . . . . . . 13 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) → 𝑟 (2ndB))
3837ad3antrrr 461 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝑟 (2ndB))
39 elprnqu 6464 . . . . . . . . . . . 12 ((⟨(1stB), (2ndB)⟩ P 𝑟 (2ndB)) → 𝑟 Q)
4036, 38, 39syl2anc 391 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝑟 Q)
41 addclnq 6359 . . . . . . . . . . 11 (((u +Q 𝑡) Q 𝑟 Q) → ((u +Q 𝑡) +Q 𝑟) Q)
4235, 40, 41syl2anc 391 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ((u +Q 𝑡) +Q 𝑟) Q)
43 prdisj 6474 . . . . . . . . . 10 ((⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P ((u +Q 𝑡) +Q 𝑟) Q) → ¬ (((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P B)) ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B))))
4428, 42, 43syl2anc 391 . . . . . . . . 9 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ¬ (((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P B)) ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B))))
45 addassnqg 6366 . . . . . . . . . . . . . . 15 ((u Q 𝑡 Q 𝑟 Q) → ((u +Q 𝑡) +Q 𝑟) = (u +Q (𝑡 +Q 𝑟)))
4632, 33, 40, 45syl3anc 1134 . . . . . . . . . . . . . 14 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ((u +Q 𝑡) +Q 𝑟) = (u +Q (𝑡 +Q 𝑟)))
47 addcomnqg 6365 . . . . . . . . . . . . . . . 16 ((𝑡 Q 𝑟 Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
4847oveq2d 5471 . . . . . . . . . . . . . . 15 ((𝑡 Q 𝑟 Q) → (u +Q (𝑡 +Q 𝑟)) = (u +Q (𝑟 +Q 𝑡)))
4933, 40, 48syl2anc 391 . . . . . . . . . . . . . 14 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (u +Q (𝑡 +Q 𝑟)) = (u +Q (𝑟 +Q 𝑡)))
5046, 49eqtrd 2069 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ((u +Q 𝑡) +Q 𝑟) = (u +Q (𝑟 +Q 𝑡)))
5150adantr 261 . . . . . . . . . . . 12 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → ((u +Q 𝑡) +Q 𝑟) = (u +Q (𝑟 +Q 𝑡)))
52 simplrl 487 . . . . . . . . . . . . 13 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → u (1stA))
53 simpr 103 . . . . . . . . . . . . 13 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → (𝑟 +Q 𝑡) (1st𝐶))
5423adantr 261 . . . . . . . . . . . . . 14 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → A P)
5522simp3d 917 . . . . . . . . . . . . . . 15 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝐶 P)
5655adantr 261 . . . . . . . . . . . . . 14 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → 𝐶 P)
57 df-iplp 6450 . . . . . . . . . . . . . . 15 +P = (𝑞 P, 𝑠 P ↦ ⟨{f Qg Q Q (g (1st𝑞) (1st𝑠) f = (g +Q ))}, {f Qg Q Q (g (2nd𝑞) (2nd𝑠) f = (g +Q ))}⟩)
58 addclnq 6359 . . . . . . . . . . . . . . 15 ((g Q Q) → (g +Q ) Q)
5957, 58genpprecll 6496 . . . . . . . . . . . . . 14 ((A P 𝐶 P) → ((u (1stA) (𝑟 +Q 𝑡) (1st𝐶)) → (u +Q (𝑟 +Q 𝑡)) (1st ‘(A +P 𝐶))))
6054, 56, 59syl2anc 391 . . . . . . . . . . . . 13 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → ((u (1stA) (𝑟 +Q 𝑡) (1st𝐶)) → (u +Q (𝑟 +Q 𝑡)) (1st ‘(A +P 𝐶))))
6152, 53, 60mp2and 409 . . . . . . . . . . . 12 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → (u +Q (𝑟 +Q 𝑡)) (1st ‘(A +P 𝐶)))
6251, 61eqeltrd 2111 . . . . . . . . . . 11 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → ((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P 𝐶)))
63 fveq2 5121 . . . . . . . . . . . . 13 ((A +P B) = (A +P 𝐶) → (1st ‘(A +P B)) = (1st ‘(A +P 𝐶)))
6463eleq2d 2104 . . . . . . . . . . . 12 ((A +P B) = (A +P 𝐶) → (((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P B)) ↔ ((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P 𝐶))))
6564ad7antlr 470 . . . . . . . . . . 11 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → (((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P B)) ↔ ((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P 𝐶))))
6662, 65mpbird 156 . . . . . . . . . 10 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → ((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P B)))
6757, 58genppreclu 6497 . . . . . . . . . . . . . . . . . . 19 ((A P B P) → (((u +Q 𝑡) (2ndA) 𝑟 (2ndB)) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B))))
6867ancomsd 256 . . . . . . . . . . . . . . . . . 18 ((A P B P) → ((𝑟 (2ndB) (u +Q 𝑡) (2ndA)) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B))))
69683adant3 923 . . . . . . . . . . . . . . . . 17 ((A P B P 𝐶 P) → ((𝑟 (2ndB) (u +Q 𝑡) (2ndA)) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B))))
7069ad2antrr 457 . . . . . . . . . . . . . . . 16 ((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) → ((𝑟 (2ndB) (u +Q 𝑡) (2ndA)) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B))))
7170imp 115 . . . . . . . . . . . . . . 15 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) (u +Q 𝑡) (2ndA))) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B)))
7271adantrlr 454 . . . . . . . . . . . . . 14 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) ((𝑟 (2ndB) 𝑟 <Q v) (u +Q 𝑡) (2ndA))) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B)))
7372anassrs 380 . . . . . . . . . . . . 13 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (u +Q 𝑡) (2ndA)) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B)))
7473ad2ant2rl 480 . . . . . . . . . . . 12 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (u (1stA) (u +Q 𝑡) (2ndA))) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B)))
7574adantlr 446 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B)))
7675adantr 261 . . . . . . . . . 10 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B)))
7766, 76jca 290 . . . . . . . . 9 (((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) (𝑟 +Q 𝑡) (1st𝐶)) → (((u +Q 𝑡) +Q 𝑟) (1st ‘(A +P B)) ((u +Q 𝑡) +Q 𝑟) (2nd ‘(A +P B))))
7844, 77mtand 590 . . . . . . . 8 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ¬ (𝑟 +Q 𝑡) (1st𝐶))
79 prop 6457 . . . . . . . . . . 11 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
8055, 79syl 14 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ⟨(1st𝐶), (2nd𝐶)⟩ P)
81 ltaddnq 6390 . . . . . . . . . . . . . 14 ((𝑡 Q 𝑡 Q) → 𝑡 <Q (𝑡 +Q 𝑡))
8233, 33, 81syl2anc 391 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝑡 <Q (𝑡 +Q 𝑡))
83 simplrr 488 . . . . . . . . . . . . 13 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (𝑡 +Q 𝑡) = w)
8482, 83breqtrd 3779 . . . . . . . . . . . 12 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → 𝑡 <Q w)
85 ltanqi 6386 . . . . . . . . . . . 12 ((𝑡 <Q w 𝑟 Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q w))
8684, 40, 85syl2anc 391 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (𝑟 +Q 𝑡) <Q (𝑟 +Q w))
87 simprr 484 . . . . . . . . . . . 12 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) → (𝑟 +Q w) = v)
8887ad2antrr 457 . . . . . . . . . . 11 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (𝑟 +Q w) = v)
8986, 88breqtrd 3779 . . . . . . . . . 10 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (𝑟 +Q 𝑡) <Q v)
90 prloc 6473 . . . . . . . . . 10 ((⟨(1st𝐶), (2nd𝐶)⟩ P (𝑟 +Q 𝑡) <Q v) → ((𝑟 +Q 𝑡) (1st𝐶) v (2nd𝐶)))
9180, 89, 90syl2anc 391 . . . . . . . . 9 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → ((𝑟 +Q 𝑡) (1st𝐶) v (2nd𝐶)))
9291orcomd 647 . . . . . . . 8 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → (v (2nd𝐶) (𝑟 +Q 𝑡) (1st𝐶)))
9378, 92ecased 1238 . . . . . . 7 ((((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) (u (1stA) (u +Q 𝑡) (2ndA))) → v (2nd𝐶))
9420, 93rexlimddv 2431 . . . . . 6 (((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) (𝑡 Q (𝑡 +Q 𝑡) = w)) → v (2nd𝐶))
9511, 94rexlimddv 2431 . . . . 5 ((((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) (w Q (𝑟 +Q w) = v)) → v (2nd𝐶))
968, 95rexlimddv 2431 . . . 4 (((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) (𝑟 (2ndB) 𝑟 <Q v)) → v (2nd𝐶))
975, 96rexlimddv 2431 . . 3 ((((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) v (2ndB)) → v (2nd𝐶))
9897ex 108 . 2 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (v (2ndB) → v (2nd𝐶)))
9998ssrdv 2945 1 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2ndB) ⊆ (2nd𝐶))
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
  Copyright terms: Public domain W3C validator