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Theorem ltexnqq 7357
Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.)
Assertion
Ref Expression
ltexnqq ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnqq
Dummy variables 𝑓 𝑔 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nqqs 7297 . . 3 Q = ((N × N) / ~Q )
2 breq1 3990 . . . 4 ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ))
3 oveq1 5857 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = (𝐴 +Q 𝑥))
43eqeq1d 2179 . . . . 5 ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
54rexbidv 2471 . . . 4 ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ∃𝑥Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
62, 5imbi12d 233 . . 3 ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ) ↔ (𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q )))
7 breq2 3991 . . . 4 ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q𝐴 <Q 𝐵))
8 eqeq2 2180 . . . . 5 ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → ((𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = 𝐵))
98rexbidv 2471 . . . 4 ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (∃𝑥Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵))
107, 9imbi12d 233 . . 3 ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → ((𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ) ↔ (𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)))
11 ordpipqqs 7323 . . . 4 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q ↔ (𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤)))
12 mulclpi 7277 . . . . . . . . 9 ((𝑦N𝑣N) → (𝑦 ·N 𝑣) ∈ N)
13 mulclpi 7277 . . . . . . . . 9 ((𝑧N𝑤N) → (𝑧 ·N 𝑤) ∈ N)
1412, 13anim12i 336 . . . . . . . 8 (((𝑦N𝑣N) ∧ (𝑧N𝑤N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
1514an42s 584 . . . . . . 7 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
16 ltexpi 7286 . . . . . . 7 (((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) ↔ ∃𝑢N ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
1715, 16syl 14 . . . . . 6 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) ↔ ∃𝑢N ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
18 df-rex 2454 . . . . . 6 (∃𝑢N ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤) ↔ ∃𝑢(𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
1917, 18bitrdi 195 . . . . 5 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) ↔ ∃𝑢(𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))))
20 simpll 524 . . . . . . . . . . . 12 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → (𝑦N𝑧N))
21 simpr 109 . . . . . . . . . . . 12 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → 𝑢N)
22 simpr 109 . . . . . . . . . . . . . . 15 ((𝑦N𝑧N) → 𝑧N)
23 simpr 109 . . . . . . . . . . . . . . 15 ((𝑤N𝑣N) → 𝑣N)
2422, 23anim12i 336 . . . . . . . . . . . . . 14 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → (𝑧N𝑣N))
2524adantr 274 . . . . . . . . . . . . 13 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → (𝑧N𝑣N))
26 mulclpi 7277 . . . . . . . . . . . . 13 ((𝑧N𝑣N) → (𝑧 ·N 𝑣) ∈ N)
2725, 26syl 14 . . . . . . . . . . . 12 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → (𝑧 ·N 𝑣) ∈ N)
2820, 21, 27jca32 308 . . . . . . . . . . 11 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → ((𝑦N𝑧N) ∧ (𝑢N ∧ (𝑧 ·N 𝑣) ∈ N)))
2928adantrr 476 . . . . . . . . . 10 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑦N𝑧N) ∧ (𝑢N ∧ (𝑧 ·N 𝑣) ∈ N)))
30 addpipqqs 7319 . . . . . . . . . 10 (((𝑦N𝑧N) ∧ (𝑢N ∧ (𝑧 ·N 𝑣) ∈ N)) → ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q )
3129, 30syl 14 . . . . . . . . 9 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q )
32 simplll 528 . . . . . . . . . . . . . . 15 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑦N)
33 simpllr 529 . . . . . . . . . . . . . . 15 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑧N)
34 simplrr 531 . . . . . . . . . . . . . . 15 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑣N)
35 mulcompig 7280 . . . . . . . . . . . . . . . 16 ((𝑓N𝑔N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
3635adantl 275 . . . . . . . . . . . . . . 15 (((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) ∧ (𝑓N𝑔N)) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
37 mulasspig 7281 . . . . . . . . . . . . . . . 16 ((𝑓N𝑔NN) → ((𝑓 ·N 𝑔) ·N ) = (𝑓 ·N (𝑔 ·N )))
3837adantl 275 . . . . . . . . . . . . . . 15 (((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) ∧ (𝑓N𝑔NN)) → ((𝑓 ·N 𝑔) ·N ) = (𝑓 ·N (𝑔 ·N )))
3932, 33, 34, 36, 38caov12d 6031 . . . . . . . . . . . . . 14 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑦 ·N (𝑧 ·N 𝑣)) = (𝑧 ·N (𝑦 ·N 𝑣)))
4039oveq1d 5865 . . . . . . . . . . . . 13 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)) = ((𝑧 ·N (𝑦 ·N 𝑣)) +N (𝑧 ·N 𝑢)))
4132, 34, 12syl2anc 409 . . . . . . . . . . . . . 14 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑦 ·N 𝑣) ∈ N)
42 simprl 526 . . . . . . . . . . . . . 14 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑢N)
43 distrpig 7282 . . . . . . . . . . . . . 14 ((𝑧N ∧ (𝑦 ·N 𝑣) ∈ N𝑢N) → (𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)) = ((𝑧 ·N (𝑦 ·N 𝑣)) +N (𝑧 ·N 𝑢)))
4433, 41, 42, 43syl3anc 1233 . . . . . . . . . . . . 13 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)) = ((𝑧 ·N (𝑦 ·N 𝑣)) +N (𝑧 ·N 𝑢)))
4540, 44eqtr4d 2206 . . . . . . . . . . . 12 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)) = (𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)))
4645opeq1d 3769 . . . . . . . . . . 11 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩ = ⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩)
4746eceq1d 6545 . . . . . . . . . 10 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q )
48 simpllr 529 . . . . . . . . . . . . 13 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → 𝑧N)
4912ad2ant2rl 508 . . . . . . . . . . . . . 14 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → (𝑦 ·N 𝑣) ∈ N)
50 addclpi 7276 . . . . . . . . . . . . . 14 (((𝑦 ·N 𝑣) ∈ N𝑢N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
5149, 50sylan 281 . . . . . . . . . . . . 13 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
5248, 51, 273jca 1172 . . . . . . . . . . . 12 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ 𝑢N) → (𝑧N ∧ ((𝑦 ·N 𝑣) +N 𝑢) ∈ N ∧ (𝑧 ·N 𝑣) ∈ N))
5352adantrr 476 . . . . . . . . . . 11 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑧N ∧ ((𝑦 ·N 𝑣) +N 𝑢) ∈ N ∧ (𝑧 ·N 𝑣) ∈ N))
54 mulcanenqec 7335 . . . . . . . . . . 11 ((𝑧N ∧ ((𝑦 ·N 𝑣) +N 𝑢) ∈ N ∧ (𝑧 ·N 𝑣) ∈ N) → [⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q )
5553, 54syl 14 . . . . . . . . . 10 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q )
5647, 55eqtrd 2203 . . . . . . . . 9 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q )
57 3anass 977 . . . . . . . . . . . . . 14 ((𝑧N𝑤N𝑣N) ↔ (𝑧N ∧ (𝑤N𝑣N)))
5857biimpri 132 . . . . . . . . . . . . 13 ((𝑧N ∧ (𝑤N𝑣N)) → (𝑧N𝑤N𝑣N))
5958adantll 473 . . . . . . . . . . . 12 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → (𝑧N𝑤N𝑣N))
6059anim1i 338 . . . . . . . . . . 11 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → ((𝑧N𝑤N𝑣N) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
6160adantrl 475 . . . . . . . . . 10 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑧N𝑤N𝑣N) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
62 opeq1 3763 . . . . . . . . . . . 12 (((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤) → ⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩ = ⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩)
6362eceq1d 6545 . . . . . . . . . . 11 (((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤) → [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q = [⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩] ~Q )
64 mulcanenqec 7335 . . . . . . . . . . 11 ((𝑧N𝑤N𝑣N) → [⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
6563, 64sylan9eqr 2225 . . . . . . . . . 10 (((𝑧N𝑤N𝑣N) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
6661, 65syl 14 . . . . . . . . 9 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
6731, 56, 663eqtrd 2207 . . . . . . . 8 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q )
6833, 34, 26syl2anc 409 . . . . . . . . . . 11 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑧 ·N 𝑣) ∈ N)
69 opelxpi 4641 . . . . . . . . . . . 12 ((𝑢N ∧ (𝑧 ·N 𝑣) ∈ N) → ⟨𝑢, (𝑧 ·N 𝑣)⟩ ∈ (N × N))
70 enqex 7309 . . . . . . . . . . . . 13 ~Q ∈ V
7170ecelqsi 6563 . . . . . . . . . . . 12 (⟨𝑢, (𝑧 ·N 𝑣)⟩ ∈ (N × N) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ∈ ((N × N) / ~Q ))
7269, 71syl 14 . . . . . . . . . . 11 ((𝑢N ∧ (𝑧 ·N 𝑣) ∈ N) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ∈ ((N × N) / ~Q ))
7342, 68, 72syl2anc 409 . . . . . . . . . 10 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ∈ ((N × N) / ~Q ))
7473, 1eleqtrrdi 2264 . . . . . . . . 9 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~QQ)
75 oveq2 5858 . . . . . . . . . . 11 (𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ))
7675eqeq1d 2179 . . . . . . . . . 10 (𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q ))
7776adantl 275 . . . . . . . . 9 (((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) ∧ 𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q ))
7874, 77rspcedv 2838 . . . . . . . 8 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
7967, 78mpd 13 . . . . . . 7 ((((𝑦N𝑧N) ∧ (𝑤N𝑣N)) ∧ (𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q )
8079ex 114 . . . . . 6 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → ((𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → ∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
8180exlimdv 1812 . . . . 5 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → (∃𝑢(𝑢N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → ∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
8219, 81sylbid 149 . . . 4 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) → ∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
8311, 82sylbid 149 . . 3 (((𝑦N𝑧N) ∧ (𝑤N𝑣N)) → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
841, 6, 10, 832ecoptocl 6597 . 2 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵))
85 ltaddnq 7356 . . . . 5 ((𝐴Q𝑥Q) → 𝐴 <Q (𝐴 +Q 𝑥))
86 breq2 3991 . . . . 5 ((𝐴 +Q 𝑥) = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑥) ↔ 𝐴 <Q 𝐵))
8785, 86syl5ibcom 154 . . . 4 ((𝐴Q𝑥Q) → ((𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
8887rexlimdva 2587 . . 3 (𝐴Q → (∃𝑥Q (𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
8988adantr 274 . 2 ((𝐴Q𝐵Q) → (∃𝑥Q (𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
9084, 89impbid 128 1 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wex 1485  wcel 2141  wrex 2449  cop 3584   class class class wbr 3987   × cxp 4607  (class class class)co 5850  [cec 6507   / cqs 6508  Ncnpi 7221   +N cpli 7222   ·N cmi 7223   <N clti 7224   ~Q ceq 7228  Qcnq 7229   +Q cplq 7231   <Q cltq 7234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-ltnqqs 7302
This theorem is referenced by:  ltexnqi  7358  addlocpr  7485  ltexprlemopl  7550  ltexprlemopu  7552  ltexprlemrl  7559  ltexprlemru  7561  cauappcvgprlemopl  7595  caucvgprlemopl  7618  caucvgprprlemopl  7646
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