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Theorem mulsproplem7 28149
Description: Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025.)
Hypotheses
Ref Expression
mulsproplem.1 (𝜑 → ∀𝑎 No 𝑏 No 𝑐 No 𝑑 No 𝑒 No 𝑓 No (((( bday 𝑎) +no ( bday 𝑏)) ∪ (((( bday 𝑐) +no ( bday 𝑒)) ∪ (( bday 𝑑) +no ( bday 𝑓))) ∪ ((( bday 𝑐) +no ( bday 𝑓)) ∪ (( bday 𝑑) +no ( bday 𝑒))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
mulsproplem7.1 (𝜑𝐴 No )
mulsproplem7.2 (𝜑𝐵 No )
mulsproplem7.3 (𝜑𝑅 ∈ ( R ‘𝐴))
mulsproplem7.4 (𝜑𝑆 ∈ ( R ‘𝐵))
mulsproplem7.5 (𝜑𝑇 ∈ ( L ‘𝐴))
mulsproplem7.6 (𝜑𝑈 ∈ ( R ‘𝐵))
Assertion
Ref Expression
mulsproplem7 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐷,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐸,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐹,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑅,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑆,𝑏,𝑐,𝑑,𝑒,𝑓   𝑇,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑈,𝑏,𝑐,𝑑,𝑒,𝑓
Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑎)   𝑈(𝑎)

Proof of Theorem mulsproplem7
StepHypRef Expression
1 rightssno 27921 . . . 4 ( R ‘𝐵) ⊆ No
2 mulsproplem7.4 . . . 4 (𝜑𝑆 ∈ ( R ‘𝐵))
31, 2sselid 3980 . . 3 (𝜑𝑆 No )
4 mulsproplem7.6 . . . 4 (𝜑𝑈 ∈ ( R ‘𝐵))
51, 4sselid 3980 . . 3 (𝜑𝑈 No )
6 sltlin 27795 . . 3 ((𝑆 No 𝑈 No ) → (𝑆 <s 𝑈𝑆 = 𝑈𝑈 <s 𝑆))
73, 5, 6syl2anc 584 . 2 (𝜑 → (𝑆 <s 𝑈𝑆 = 𝑈𝑈 <s 𝑆))
8 mulsproplem.1 . . . . . . . . 9 (𝜑 → ∀𝑎 No 𝑏 No 𝑐 No 𝑑 No 𝑒 No 𝑓 No (((( bday 𝑎) +no ( bday 𝑏)) ∪ (((( bday 𝑐) +no ( bday 𝑒)) ∪ (( bday 𝑑) +no ( bday 𝑓))) ∪ ((( bday 𝑐) +no ( bday 𝑓)) ∪ (( bday 𝑑) +no ( bday 𝑒))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
9 rightssold 27919 . . . . . . . . . 10 ( R ‘𝐴) ⊆ ( O ‘( bday 𝐴))
10 mulsproplem7.3 . . . . . . . . . 10 (𝜑𝑅 ∈ ( R ‘𝐴))
119, 10sselid 3980 . . . . . . . . 9 (𝜑𝑅 ∈ ( O ‘( bday 𝐴)))
12 mulsproplem7.2 . . . . . . . . 9 (𝜑𝐵 No )
138, 11, 12mulsproplem2 28144 . . . . . . . 8 (𝜑 → (𝑅 ·s 𝐵) ∈ No )
14 mulsproplem7.1 . . . . . . . . 9 (𝜑𝐴 No )
15 rightssold 27919 . . . . . . . . . 10 ( R ‘𝐵) ⊆ ( O ‘( bday 𝐵))
1615, 2sselid 3980 . . . . . . . . 9 (𝜑𝑆 ∈ ( O ‘( bday 𝐵)))
178, 14, 16mulsproplem3 28145 . . . . . . . 8 (𝜑 → (𝐴 ·s 𝑆) ∈ No )
1813, 17addscld 28014 . . . . . . 7 (𝜑 → ((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) ∈ No )
198, 11, 16mulsproplem4 28146 . . . . . . 7 (𝜑 → (𝑅 ·s 𝑆) ∈ No )
2018, 19subscld 28094 . . . . . 6 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) ∈ No )
2120adantr 480 . . . . 5 ((𝜑𝑆 <s 𝑈) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) ∈ No )
22 leftssold 27918 . . . . . . . . . 10 ( L ‘𝐴) ⊆ ( O ‘( bday 𝐴))
23 mulsproplem7.5 . . . . . . . . . 10 (𝜑𝑇 ∈ ( L ‘𝐴))
2422, 23sselid 3980 . . . . . . . . 9 (𝜑𝑇 ∈ ( O ‘( bday 𝐴)))
258, 24, 12mulsproplem2 28144 . . . . . . . 8 (𝜑 → (𝑇 ·s 𝐵) ∈ No )
2625, 17addscld 28014 . . . . . . 7 (𝜑 → ((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) ∈ No )
278, 24, 16mulsproplem4 28146 . . . . . . 7 (𝜑 → (𝑇 ·s 𝑆) ∈ No )
2826, 27subscld 28094 . . . . . 6 (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)) ∈ No )
2928adantr 480 . . . . 5 ((𝜑𝑆 <s 𝑈) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)) ∈ No )
3015, 4sselid 3980 . . . . . . . . 9 (𝜑𝑈 ∈ ( O ‘( bday 𝐵)))
318, 14, 30mulsproplem3 28145 . . . . . . . 8 (𝜑 → (𝐴 ·s 𝑈) ∈ No )
3225, 31addscld 28014 . . . . . . 7 (𝜑 → ((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) ∈ No )
338, 24, 30mulsproplem4 28146 . . . . . . 7 (𝜑 → (𝑇 ·s 𝑈) ∈ No )
3432, 33subscld 28094 . . . . . 6 (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) ∈ No )
3534adantr 480 . . . . 5 ((𝜑𝑆 <s 𝑈) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) ∈ No )
36 lltropt 27912 . . . . . . . . . . 11 ( L ‘𝐴) <<s ( R ‘𝐴)
3736a1i 11 . . . . . . . . . 10 (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
3837, 23, 10ssltsepcd 27840 . . . . . . . . 9 (𝜑𝑇 <s 𝑅)
39 ssltright 27911 . . . . . . . . . . 11 (𝐵 No → {𝐵} <<s ( R ‘𝐵))
4012, 39syl 17 . . . . . . . . . 10 (𝜑 → {𝐵} <<s ( R ‘𝐵))
41 snidg 4659 . . . . . . . . . . 11 (𝐵 No 𝐵 ∈ {𝐵})
4212, 41syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ {𝐵})
4340, 42, 2ssltsepcd 27840 . . . . . . . . 9 (𝜑𝐵 <s 𝑆)
44 0sno 27872 . . . . . . . . . . . 12 0s No
4544a1i 11 . . . . . . . . . . 11 (𝜑 → 0s No )
46 leftssno 27920 . . . . . . . . . . . 12 ( L ‘𝐴) ⊆ No
4746, 23sselid 3980 . . . . . . . . . . 11 (𝜑𝑇 No )
48 rightssno 27921 . . . . . . . . . . . 12 ( R ‘𝐴) ⊆ No
4948, 10sselid 3980 . . . . . . . . . . 11 (𝜑𝑅 No )
50 bday0s 27874 . . . . . . . . . . . . . . . 16 ( bday ‘ 0s ) = ∅
5150, 50oveq12i 7444 . . . . . . . . . . . . . . 15 (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
52 0elon 6437 . . . . . . . . . . . . . . . 16 ∅ ∈ On
53 naddrid 8722 . . . . . . . . . . . . . . . 16 (∅ ∈ On → (∅ +no ∅) = ∅)
5452, 53ax-mp 5 . . . . . . . . . . . . . . 15 (∅ +no ∅) = ∅
5551, 54eqtri 2764 . . . . . . . . . . . . . 14 (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
5655uneq1i 4163 . . . . . . . . . . . . 13 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) = (∅ ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))))
57 0un 4395 . . . . . . . . . . . . 13 (∅ ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) = (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))))
5856, 57eqtri 2764 . . . . . . . . . . . 12 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) = (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))))
59 oldbdayim 27928 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑇) ∈ ( bday 𝐴))
6024, 59syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ( bday 𝑇) ∈ ( bday 𝐴))
61 bdayelon 27822 . . . . . . . . . . . . . . . . 17 ( bday 𝑇) ∈ On
62 bdayelon 27822 . . . . . . . . . . . . . . . . 17 ( bday 𝐴) ∈ On
63 bdayelon 27822 . . . . . . . . . . . . . . . . 17 ( bday 𝐵) ∈ On
64 naddel1 8726 . . . . . . . . . . . . . . . . 17 ((( bday 𝑇) ∈ On ∧ ( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → (( bday 𝑇) ∈ ( bday 𝐴) ↔ (( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
6561, 62, 63, 64mp3an 1462 . . . . . . . . . . . . . . . 16 (( bday 𝑇) ∈ ( bday 𝐴) ↔ (( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
6660, 65sylib 218 . . . . . . . . . . . . . . 15 (𝜑 → (( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
67 oldbdayim 27928 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑅) ∈ ( bday 𝐴))
6811, 67syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ( bday 𝑅) ∈ ( bday 𝐴))
69 oldbdayim 27928 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ( O ‘( bday 𝐵)) → ( bday 𝑆) ∈ ( bday 𝐵))
7016, 69syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ( bday 𝑆) ∈ ( bday 𝐵))
71 naddel12 8739 . . . . . . . . . . . . . . . . 17 ((( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → ((( bday 𝑅) ∈ ( bday 𝐴) ∧ ( bday 𝑆) ∈ ( bday 𝐵)) → (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
7262, 63, 71mp2an 692 . . . . . . . . . . . . . . . 16 ((( bday 𝑅) ∈ ( bday 𝐴) ∧ ( bday 𝑆) ∈ ( bday 𝐵)) → (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
7368, 70, 72syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
7466, 73jca 511 . . . . . . . . . . . . . 14 (𝜑 → ((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
75 naddel12 8739 . . . . . . . . . . . . . . . . 17 ((( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → ((( bday 𝑇) ∈ ( bday 𝐴) ∧ ( bday 𝑆) ∈ ( bday 𝐵)) → (( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
7662, 63, 75mp2an 692 . . . . . . . . . . . . . . . 16 ((( bday 𝑇) ∈ ( bday 𝐴) ∧ ( bday 𝑆) ∈ ( bday 𝐵)) → (( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
7760, 70, 76syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
78 bdayelon 27822 . . . . . . . . . . . . . . . . 17 ( bday 𝑅) ∈ On
79 naddel1 8726 . . . . . . . . . . . . . . . . 17 ((( bday 𝑅) ∈ On ∧ ( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → (( bday 𝑅) ∈ ( bday 𝐴) ↔ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
8078, 62, 63, 79mp3an 1462 . . . . . . . . . . . . . . . 16 (( bday 𝑅) ∈ ( bday 𝐴) ↔ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
8168, 80sylib 218 . . . . . . . . . . . . . . 15 (𝜑 → (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
8277, 81jca 511 . . . . . . . . . . . . . 14 (𝜑 → ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
83 naddcl 8716 . . . . . . . . . . . . . . . . . 18 ((( bday 𝑇) ∈ On ∧ ( bday 𝐵) ∈ On) → (( bday 𝑇) +no ( bday 𝐵)) ∈ On)
8461, 63, 83mp2an 692 . . . . . . . . . . . . . . . . 17 (( bday 𝑇) +no ( bday 𝐵)) ∈ On
85 bdayelon 27822 . . . . . . . . . . . . . . . . . 18 ( bday 𝑆) ∈ On
86 naddcl 8716 . . . . . . . . . . . . . . . . . 18 ((( bday 𝑅) ∈ On ∧ ( bday 𝑆) ∈ On) → (( bday 𝑅) +no ( bday 𝑆)) ∈ On)
8778, 85, 86mp2an 692 . . . . . . . . . . . . . . . . 17 (( bday 𝑅) +no ( bday 𝑆)) ∈ On
8884, 87onun2i 6505 . . . . . . . . . . . . . . . 16 ((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ On
89 naddcl 8716 . . . . . . . . . . . . . . . . . 18 ((( bday 𝑇) ∈ On ∧ ( bday 𝑆) ∈ On) → (( bday 𝑇) +no ( bday 𝑆)) ∈ On)
9061, 85, 89mp2an 692 . . . . . . . . . . . . . . . . 17 (( bday 𝑇) +no ( bday 𝑆)) ∈ On
91 naddcl 8716 . . . . . . . . . . . . . . . . . 18 ((( bday 𝑅) ∈ On ∧ ( bday 𝐵) ∈ On) → (( bday 𝑅) +no ( bday 𝐵)) ∈ On)
9278, 63, 91mp2an 692 . . . . . . . . . . . . . . . . 17 (( bday 𝑅) +no ( bday 𝐵)) ∈ On
9390, 92onun2i 6505 . . . . . . . . . . . . . . . 16 ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ On
94 naddcl 8716 . . . . . . . . . . . . . . . . 17 ((( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → (( bday 𝐴) +no ( bday 𝐵)) ∈ On)
9562, 63, 94mp2an 692 . . . . . . . . . . . . . . . 16 (( bday 𝐴) +no ( bday 𝐵)) ∈ On
96 onunel 6488 . . . . . . . . . . . . . . . 16 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ On ∧ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
9788, 93, 95, 96mp3an 1462 . . . . . . . . . . . . . . 15 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵))))
98 onunel 6488 . . . . . . . . . . . . . . . . 17 (((( bday 𝑇) +no ( bday 𝐵)) ∈ On ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
9984, 87, 95, 98mp3an 1462 . . . . . . . . . . . . . . . 16 (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
100 onunel 6488 . . . . . . . . . . . . . . . . 17 (((( bday 𝑇) +no ( bday 𝑆)) ∈ On ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
10190, 92, 95, 100mp3an 1462 . . . . . . . . . . . . . . . 16 (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
10299, 101anbi12i 628 . . . . . . . . . . . . . . 15 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵))) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
10397, 102bitri 275 . . . . . . . . . . . . . 14 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
10474, 82, 103sylanbrc 583 . . . . . . . . . . . . 13 (𝜑 → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)))
105 elun1 4181 . . . . . . . . . . . . 13 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
106104, 105syl 17 . . . . . . . . . . . 12 (𝜑 → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
10758, 106eqeltrid 2844 . . . . . . . . . . 11 (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
1088, 45, 45, 47, 49, 12, 3, 107mulsproplem1 28143 . . . . . . . . . 10 (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑇 <s 𝑅𝐵 <s 𝑆) → ((𝑇 ·s 𝑆) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝐵)))))
109108simprd 495 . . . . . . . . 9 (𝜑 → ((𝑇 <s 𝑅𝐵 <s 𝑆) → ((𝑇 ·s 𝑆) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝐵))))
11038, 43, 109mp2and 699 . . . . . . . 8 (𝜑 → ((𝑇 ·s 𝑆) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝐵)))
11127, 25, 19, 13sltsubsub2bd 28115 . . . . . . . . 9 (𝜑 → (((𝑇 ·s 𝑆) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝐵)) ↔ ((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑆)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑆))))
11213, 19subscld 28094 . . . . . . . . . 10 (𝜑 → ((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑆)) ∈ No )
11325, 27subscld 28094 . . . . . . . . . 10 (𝜑 → ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑆)) ∈ No )
114112, 113, 17sltadd1d 28032 . . . . . . . . 9 (𝜑 → (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑆)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑆)) ↔ (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑆)) +s (𝐴 ·s 𝑆)) <s (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑆)) +s (𝐴 ·s 𝑆))))
115111, 114bitrd 279 . . . . . . . 8 (𝜑 → (((𝑇 ·s 𝑆) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝐵)) ↔ (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑆)) +s (𝐴 ·s 𝑆)) <s (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑆)) +s (𝐴 ·s 𝑆))))
116110, 115mpbid 232 . . . . . . 7 (𝜑 → (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑆)) +s (𝐴 ·s 𝑆)) <s (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑆)) +s (𝐴 ·s 𝑆)))
11713, 17, 19addsubsd 28113 . . . . . . 7 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) = (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑆)) +s (𝐴 ·s 𝑆)))
11825, 17, 27addsubsd 28113 . . . . . . 7 (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)) = (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑆)) +s (𝐴 ·s 𝑆)))
119116, 117, 1183brtr4d 5174 . . . . . 6 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)))
120119adantr 480 . . . . 5 ((𝜑𝑆 <s 𝑈) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)))
121 ssltleft 27910 . . . . . . . . . . 11 (𝐴 No → ( L ‘𝐴) <<s {𝐴})
12214, 121syl 17 . . . . . . . . . 10 (𝜑 → ( L ‘𝐴) <<s {𝐴})
123 snidg 4659 . . . . . . . . . . 11 (𝐴 No 𝐴 ∈ {𝐴})
12414, 123syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
125122, 23, 124ssltsepcd 27840 . . . . . . . . 9 (𝜑𝑇 <s 𝐴)
12655uneq1i 4163 . . . . . . . . . . . . 13 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))))) = (∅ ∪ (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))))
127 0un 4395 . . . . . . . . . . . . 13 (∅ ∪ (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))))) = (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))))
128126, 127eqtri 2764 . . . . . . . . . . . 12 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))))) = (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))))
129 oldbdayim 27928 . . . . . . . . . . . . . . . . 17 (𝑈 ∈ ( O ‘( bday 𝐵)) → ( bday 𝑈) ∈ ( bday 𝐵))
13030, 129syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ( bday 𝑈) ∈ ( bday 𝐵))
131 bdayelon 27822 . . . . . . . . . . . . . . . . 17 ( bday 𝑈) ∈ On
132 naddel2 8727 . . . . . . . . . . . . . . . . 17 ((( bday 𝑈) ∈ On ∧ ( bday 𝐵) ∈ On ∧ ( bday 𝐴) ∈ On) → (( bday 𝑈) ∈ ( bday 𝐵) ↔ (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
133131, 63, 62, 132mp3an 1462 . . . . . . . . . . . . . . . 16 (( bday 𝑈) ∈ ( bday 𝐵) ↔ (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
134130, 133sylib 218 . . . . . . . . . . . . . . 15 (𝜑 → (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
13577, 134jca 511 . . . . . . . . . . . . . 14 (𝜑 → ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
136 naddel12 8739 . . . . . . . . . . . . . . . . 17 ((( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → ((( bday 𝑇) ∈ ( bday 𝐴) ∧ ( bday 𝑈) ∈ ( bday 𝐵)) → (( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
13762, 63, 136mp2an 692 . . . . . . . . . . . . . . . 16 ((( bday 𝑇) ∈ ( bday 𝐴) ∧ ( bday 𝑈) ∈ ( bday 𝐵)) → (( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
13860, 130, 137syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
139 naddel2 8727 . . . . . . . . . . . . . . . . 17 ((( bday 𝑆) ∈ On ∧ ( bday 𝐵) ∈ On ∧ ( bday 𝐴) ∈ On) → (( bday 𝑆) ∈ ( bday 𝐵) ↔ (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
14085, 63, 62, 139mp3an 1462 . . . . . . . . . . . . . . . 16 (( bday 𝑆) ∈ ( bday 𝐵) ↔ (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
14170, 140sylib 218 . . . . . . . . . . . . . . 15 (𝜑 → (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
142138, 141jca 511 . . . . . . . . . . . . . 14 (𝜑 → ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
143 naddcl 8716 . . . . . . . . . . . . . . . . . 18 ((( bday 𝐴) ∈ On ∧ ( bday 𝑈) ∈ On) → (( bday 𝐴) +no ( bday 𝑈)) ∈ On)
14462, 131, 143mp2an 692 . . . . . . . . . . . . . . . . 17 (( bday 𝐴) +no ( bday 𝑈)) ∈ On
14590, 144onun2i 6505 . . . . . . . . . . . . . . . 16 ((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∈ On
146 naddcl 8716 . . . . . . . . . . . . . . . . . 18 ((( bday 𝑇) ∈ On ∧ ( bday 𝑈) ∈ On) → (( bday 𝑇) +no ( bday 𝑈)) ∈ On)
14761, 131, 146mp2an 692 . . . . . . . . . . . . . . . . 17 (( bday 𝑇) +no ( bday 𝑈)) ∈ On
148 naddcl 8716 . . . . . . . . . . . . . . . . . 18 ((( bday 𝐴) ∈ On ∧ ( bday 𝑆) ∈ On) → (( bday 𝐴) +no ( bday 𝑆)) ∈ On)
14962, 85, 148mp2an 692 . . . . . . . . . . . . . . . . 17 (( bday 𝐴) +no ( bday 𝑆)) ∈ On
150147, 149onun2i 6505 . . . . . . . . . . . . . . . 16 ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))) ∈ On
151 onunel 6488 . . . . . . . . . . . . . . . 16 ((((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∈ On ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → ((((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
152145, 150, 95, 151mp3an 1462 . . . . . . . . . . . . . . 15 ((((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵))))
153 onunel 6488 . . . . . . . . . . . . . . . . 17 (((( bday 𝑇) +no ( bday 𝑆)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝑈)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
15490, 144, 95, 153mp3an 1462 . . . . . . . . . . . . . . . 16 (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
155 onunel 6488 . . . . . . . . . . . . . . . . 17 (((( bday 𝑇) +no ( bday 𝑈)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝑆)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
156147, 149, 95, 155mp3an 1462 . . . . . . . . . . . . . . . 16 (((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
157154, 156anbi12i 628 . . . . . . . . . . . . . . 15 ((((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵))) ↔ (((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
158152, 157bitri 275 . . . . . . . . . . . . . 14 ((((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
159135, 142, 158sylanbrc 583 . . . . . . . . . . . . 13 (𝜑 → (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))) ∈ (( bday 𝐴) +no ( bday 𝐵)))
160 elun1 4181 . . . . . . . . . . . . 13 ((((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) → (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
161159, 160syl 17 . . . . . . . . . . . 12 (𝜑 → (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
162128, 161eqeltrid 2844 . . . . . . . . . . 11 (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝑆)) ∪ (( bday 𝐴) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝐴) +no ( bday 𝑆))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
1638, 45, 45, 47, 14, 3, 5, 162mulsproplem1 28143 . . . . . . . . . 10 (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑇 <s 𝐴𝑆 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑆)))))
164163simprd 495 . . . . . . . . 9 (𝜑 → ((𝑇 <s 𝐴𝑆 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑆))))
165125, 164mpand 695 . . . . . . . 8 (𝜑 → (𝑆 <s 𝑈 → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑆))))
166165imp 406 . . . . . . 7 ((𝜑𝑆 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑆)))
16733, 31, 27, 17sltsubsub3bd 28116 . . . . . . . . 9 (𝜑 → (((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑆)) ↔ ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈))))
16817, 27subscld 28094 . . . . . . . . . 10 (𝜑 → ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆)) ∈ No )
16931, 33subscld 28094 . . . . . . . . . 10 (𝜑 → ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈)) ∈ No )
170168, 169, 25sltadd2d 28031 . . . . . . . . 9 (𝜑 → (((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈)) ↔ ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆))) <s ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈)))))
171167, 170bitrd 279 . . . . . . . 8 (𝜑 → (((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑆)) ↔ ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆))) <s ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈)))))
172171adantr 480 . . . . . . 7 ((𝜑𝑆 <s 𝑈) → (((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑆)) ↔ ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆))) <s ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈)))))
173166, 172mpbid 232 . . . . . 6 ((𝜑𝑆 <s 𝑈) → ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆))) <s ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈))))
17425, 17, 27addsubsassd 28112 . . . . . . 7 (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)) = ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆))))
175174adantr 480 . . . . . 6 ((𝜑𝑆 <s 𝑈) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)) = ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑇 ·s 𝑆))))
17625, 31, 33addsubsassd 28112 . . . . . . 7 (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) = ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈))))
177176adantr 480 . . . . . 6 ((𝜑𝑆 <s 𝑈) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) = ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈))))
178173, 175, 1773brtr4d 5174 . . . . 5 ((𝜑𝑆 <s 𝑈) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑇 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
17921, 29, 35, 120, 178slttrd 27805 . . . 4 ((𝜑𝑆 <s 𝑈) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
180179ex 412 . . 3 (𝜑 → (𝑆 <s 𝑈 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈))))
18140, 42, 4ssltsepcd 27840 . . . . . . 7 (𝜑𝐵 <s 𝑈)
18255uneq1i 4163 . . . . . . . . . . 11 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) = (∅ ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))))
183 0un 4395 . . . . . . . . . . 11 (∅ ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) = (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))))
184182, 183eqtri 2764 . . . . . . . . . 10 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) = (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))))
185 naddel12 8739 . . . . . . . . . . . . . . 15 ((( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → ((( bday 𝑅) ∈ ( bday 𝐴) ∧ ( bday 𝑈) ∈ ( bday 𝐵)) → (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
18662, 63, 185mp2an 692 . . . . . . . . . . . . . 14 ((( bday 𝑅) ∈ ( bday 𝐴) ∧ ( bday 𝑈) ∈ ( bday 𝐵)) → (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
18768, 130, 186syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
18866, 187jca 511 . . . . . . . . . . . 12 (𝜑 → ((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
189138, 81jca 511 . . . . . . . . . . . 12 (𝜑 → ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
190 naddcl 8716 . . . . . . . . . . . . . . . 16 ((( bday 𝑅) ∈ On ∧ ( bday 𝑈) ∈ On) → (( bday 𝑅) +no ( bday 𝑈)) ∈ On)
19178, 131, 190mp2an 692 . . . . . . . . . . . . . . 15 (( bday 𝑅) +no ( bday 𝑈)) ∈ On
19284, 191onun2i 6505 . . . . . . . . . . . . . 14 ((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ On
193147, 92onun2i 6505 . . . . . . . . . . . . . 14 ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ On
194 onunel 6488 . . . . . . . . . . . . . 14 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ On ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
195192, 193, 95, 194mp3an 1462 . . . . . . . . . . . . 13 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵))))
196 onunel 6488 . . . . . . . . . . . . . . 15 (((( bday 𝑇) +no ( bday 𝐵)) ∈ On ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
19784, 191, 95, 196mp3an 1462 . . . . . . . . . . . . . 14 (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
198 onunel 6488 . . . . . . . . . . . . . . 15 (((( bday 𝑇) +no ( bday 𝑈)) ∈ On ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
199147, 92, 95, 198mp3an 1462 . . . . . . . . . . . . . 14 (((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
200197, 199anbi12i 628 . . . . . . . . . . . . 13 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))) ∈ (( bday 𝐴) +no ( bday 𝐵))) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
201195, 200bitri 275 . . . . . . . . . . . 12 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝑇) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝑇) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
202188, 189, 201sylanbrc 583 . . . . . . . . . . 11 (𝜑 → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)))
203 elun1 4181 . . . . . . . . . . 11 ((((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
204202, 203syl 17 . . . . . . . . . 10 (𝜑 → (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
205184, 204eqeltrid 2844 . . . . . . . . 9 (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝑇) +no ( bday 𝐵)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∪ ((( bday 𝑇) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝐵))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
2068, 45, 45, 47, 49, 12, 5, 205mulsproplem1 28143 . . . . . . . 8 (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑇 <s 𝑅𝐵 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑈) -s (𝑅 ·s 𝐵)))))
207206simprd 495 . . . . . . 7 (𝜑 → ((𝑇 <s 𝑅𝐵 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑈) -s (𝑅 ·s 𝐵))))
20838, 181, 207mp2and 699 . . . . . 6 (𝜑 → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑈) -s (𝑅 ·s 𝐵)))
2098, 11, 30mulsproplem4 28146 . . . . . . . 8 (𝜑 → (𝑅 ·s 𝑈) ∈ No )
21033, 25, 209, 13sltsubsub2bd 28115 . . . . . . 7 (𝜑 → (((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑈) -s (𝑅 ·s 𝐵)) ↔ ((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑈)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈))))
21113, 209subscld 28094 . . . . . . . 8 (𝜑 → ((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑈)) ∈ No )
21225, 33subscld 28094 . . . . . . . 8 (𝜑 → ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) ∈ No )
213211, 212, 31sltadd1d 28032 . . . . . . 7 (𝜑 → (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑈)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) ↔ (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑈)) +s (𝐴 ·s 𝑈)) <s (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) +s (𝐴 ·s 𝑈))))
214210, 213bitrd 279 . . . . . 6 (𝜑 → (((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑅 ·s 𝑈) -s (𝑅 ·s 𝐵)) ↔ (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑈)) +s (𝐴 ·s 𝑈)) <s (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) +s (𝐴 ·s 𝑈))))
215208, 214mpbid 232 . . . . 5 (𝜑 → (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑈)) +s (𝐴 ·s 𝑈)) <s (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) +s (𝐴 ·s 𝑈)))
21613, 31, 209addsubsd 28113 . . . . 5 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) = (((𝑅 ·s 𝐵) -s (𝑅 ·s 𝑈)) +s (𝐴 ·s 𝑈)))
21725, 31, 33addsubsd 28113 . . . . 5 (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) = (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) +s (𝐴 ·s 𝑈)))
218215, 216, 2173brtr4d 5174 . . . 4 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
219 oveq2 7440 . . . . . . 7 (𝑆 = 𝑈 → (𝐴 ·s 𝑆) = (𝐴 ·s 𝑈))
220219oveq2d 7448 . . . . . 6 (𝑆 = 𝑈 → ((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) = ((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)))
221 oveq2 7440 . . . . . 6 (𝑆 = 𝑈 → (𝑅 ·s 𝑆) = (𝑅 ·s 𝑈))
222220, 221oveq12d 7450 . . . . 5 (𝑆 = 𝑈 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) = (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)))
223222breq1d 5152 . . . 4 (𝑆 = 𝑈 → ((((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) ↔ (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈))))
224218, 223syl5ibrcom 247 . . 3 (𝜑 → (𝑆 = 𝑈 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈))))
22520adantr 480 . . . . 5 ((𝜑𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) ∈ No )
22613, 31addscld 28014 . . . . . . 7 (𝜑 → ((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) ∈ No )
227226, 209subscld 28094 . . . . . 6 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) ∈ No )
228227adantr 480 . . . . 5 ((𝜑𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) ∈ No )
22934adantr 480 . . . . 5 ((𝜑𝑈 <s 𝑆) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) ∈ No )
230 ssltright 27911 . . . . . . . . . . 11 (𝐴 No → {𝐴} <<s ( R ‘𝐴))
23114, 230syl 17 . . . . . . . . . 10 (𝜑 → {𝐴} <<s ( R ‘𝐴))
232231, 124, 10ssltsepcd 27840 . . . . . . . . 9 (𝜑𝐴 <s 𝑅)
23355uneq1i 4163 . . . . . . . . . . . . 13 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))))) = (∅ ∪ (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))))
234 0un 4395 . . . . . . . . . . . . 13 (∅ ∪ (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))))) = (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))))
235233, 234eqtri 2764 . . . . . . . . . . . 12 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))))) = (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))))
236134, 73jca 511 . . . . . . . . . . . . . 14 (𝜑 → ((( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
237141, 187jca 511 . . . . . . . . . . . . . 14 (𝜑 → ((( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
238144, 87onun2i 6505 . . . . . . . . . . . . . . . 16 ((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ On
239149, 191onun2i 6505 . . . . . . . . . . . . . . . 16 ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ On
240 onunel 6488 . . . . . . . . . . . . . . . 16 ((((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ On ∧ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → ((((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
241238, 239, 95, 240mp3an 1462 . . . . . . . . . . . . . . 15 ((((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵))))
242 onunel 6488 . . . . . . . . . . . . . . . . 17 (((( bday 𝐴) +no ( bday 𝑈)) ∈ On ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
243144, 87, 95, 242mp3an 1462 . . . . . . . . . . . . . . . 16 (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
244 onunel 6488 . . . . . . . . . . . . . . . . 17 (((( bday 𝐴) +no ( bday 𝑆)) ∈ On ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ On ∧ (( bday 𝐴) +no ( bday 𝐵)) ∈ On) → (((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
245149, 191, 95, 244mp3an 1462 . . . . . . . . . . . . . . . 16 (((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ ((( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
246243, 245anbi12i 628 . . . . . . . . . . . . . . 15 ((((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))) ∈ (( bday 𝐴) +no ( bday 𝐵))) ↔ (((( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
247241, 246bitri 275 . . . . . . . . . . . . . 14 ((((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) ↔ (((( bday 𝐴) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵))) ∧ ((( bday 𝐴) +no ( bday 𝑆)) ∈ (( bday 𝐴) +no ( bday 𝐵)) ∧ (( bday 𝑅) +no ( bday 𝑈)) ∈ (( bday 𝐴) +no ( bday 𝐵)))))
248236, 237, 247sylanbrc 583 . . . . . . . . . . . . 13 (𝜑 → (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))) ∈ (( bday 𝐴) +no ( bday 𝐵)))
249 elun1 4181 . . . . . . . . . . . . 13 ((((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))) ∈ (( bday 𝐴) +no ( bday 𝐵)) → (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
250248, 249syl 17 . . . . . . . . . . . 12 (𝜑 → (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈)))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
251235, 250eqeltrid 2844 . . . . . . . . . . 11 (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday 𝐴) +no ( bday 𝑈)) ∪ (( bday 𝑅) +no ( bday 𝑆))) ∪ ((( bday 𝐴) +no ( bday 𝑆)) ∪ (( bday 𝑅) +no ( bday 𝑈))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday 𝐶) +no ( bday 𝐸)) ∪ (( bday 𝐷) +no ( bday 𝐹))) ∪ ((( bday 𝐶) +no ( bday 𝐹)) ∪ (( bday 𝐷) +no ( bday 𝐸))))))
2528, 45, 45, 14, 49, 5, 3, 251mulsproplem1 28143 . . . . . . . . . 10 (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝐴 <s 𝑅𝑈 <s 𝑆) → ((𝐴 ·s 𝑆) -s (𝐴 ·s 𝑈)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝑈)))))
253252simprd 495 . . . . . . . . 9 (𝜑 → ((𝐴 <s 𝑅𝑈 <s 𝑆) → ((𝐴 ·s 𝑆) -s (𝐴 ·s 𝑈)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝑈))))
254232, 253mpand 695 . . . . . . . 8 (𝜑 → (𝑈 <s 𝑆 → ((𝐴 ·s 𝑆) -s (𝐴 ·s 𝑈)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝑈))))
255254imp 406 . . . . . . 7 ((𝜑𝑈 <s 𝑆) → ((𝐴 ·s 𝑆) -s (𝐴 ·s 𝑈)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝑈)))
25617, 19, 31, 209sltsubsubbd 28114 . . . . . . . . 9 (𝜑 → (((𝐴 ·s 𝑆) -s (𝐴 ·s 𝑈)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝑈)) ↔ ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈))))
25717, 19subscld 28094 . . . . . . . . . 10 (𝜑 → ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆)) ∈ No )
25831, 209subscld 28094 . . . . . . . . . 10 (𝜑 → ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈)) ∈ No )
259257, 258, 13sltadd2d 28031 . . . . . . . . 9 (𝜑 → (((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆)) <s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈)) ↔ ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆))) <s ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈)))))
260256, 259bitrd 279 . . . . . . . 8 (𝜑 → (((𝐴 ·s 𝑆) -s (𝐴 ·s 𝑈)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝑈)) ↔ ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆))) <s ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈)))))
261260adantr 480 . . . . . . 7 ((𝜑𝑈 <s 𝑆) → (((𝐴 ·s 𝑆) -s (𝐴 ·s 𝑈)) <s ((𝑅 ·s 𝑆) -s (𝑅 ·s 𝑈)) ↔ ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆))) <s ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈)))))
262255, 261mpbid 232 . . . . . 6 ((𝜑𝑈 <s 𝑆) → ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆))) <s ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈))))
26313, 17, 19addsubsassd 28112 . . . . . . 7 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) = ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆))))
264263adantr 480 . . . . . 6 ((𝜑𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) = ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑆) -s (𝑅 ·s 𝑆))))
26513, 31, 209addsubsassd 28112 . . . . . . 7 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) = ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈))))
266265adantr 480 . . . . . 6 ((𝜑𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) = ((𝑅 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑅 ·s 𝑈))))
267262, 264, 2663brtr4d 5174 . . . . 5 ((𝜑𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)))
268218adantr 480 . . . . 5 ((𝜑𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑅 ·s 𝑈)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
269225, 228, 229, 267, 268slttrd 27805 . . . 4 ((𝜑𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
270269ex 412 . . 3 (𝜑 → (𝑈 <s 𝑆 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈))))
271180, 224, 2703jaod 1430 . 2 (𝜑 → ((𝑆 <s 𝑈𝑆 = 𝑈𝑈 <s 𝑆) → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈))))
2727, 271mpd 15 1 (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3o 1085   = wceq 1539  wcel 2107  wral 3060  cun 3948  c0 4332  {csn 4625   class class class wbr 5142  Oncon0 6383  cfv 6560  (class class class)co 7432   +no cnadd 8704   No csur 27685   <s cslt 27686   bday cbday 27687   <<s csslt 27826   0s c0s 27868   O cold 27883   L cleft 27885   R cright 27886   +s cadds 27993   -s csubs 28053   ·s cmuls 28133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055
This theorem is referenced by:  mulsproplem9  28151
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