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Theorem ssltmul1 28188
Description: One surreal set less-than relationship for cuts of 𝐴 and 𝐵. (Contributed by Scott Fenton, 7-Mar-2025.)
Hypotheses
Ref Expression
ssltmul1.1 (𝜑𝐿 <<s 𝑅)
ssltmul1.2 (𝜑𝑀 <<s 𝑆)
ssltmul1.3 (𝜑𝐴 = (𝐿 |s 𝑅))
ssltmul1.4 (𝜑𝐵 = (𝑀 |s 𝑆))
Assertion
Ref Expression
ssltmul1 (𝜑 → ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)})
Distinct variable groups:   𝐴,𝑎   𝐴,𝑏   𝐴,𝑝,𝑞   𝐴,𝑟,𝑠   𝐵,𝑎   𝐵,𝑏   𝐵,𝑝,𝑞   𝐵,𝑟,𝑠   𝐿,𝑎,𝑝,𝑞   𝑀,𝑎,𝑝,𝑞   𝑅,𝑏,𝑟,𝑠   𝑆,𝑏,𝑟,𝑠   𝜑,𝑝,𝑎,𝑞   𝜑,𝑏,𝑟,𝑠
Allowed substitution hints:   𝑅(𝑞,𝑝,𝑎)   𝑆(𝑞,𝑝,𝑎)   𝐿(𝑠,𝑟,𝑏)   𝑀(𝑠,𝑟,𝑏)

Proof of Theorem ssltmul1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . . 5 (𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) = (𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
21rnmpo 7566 . . . 4 ran (𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) = {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))}
3 ssltmul1.1 . . . . . . 7 (𝜑𝐿 <<s 𝑅)
4 ssltex1 27846 . . . . . . 7 (𝐿 <<s 𝑅𝐿 ∈ V)
53, 4syl 17 . . . . . 6 (𝜑𝐿 ∈ V)
6 ssltmul1.2 . . . . . . 7 (𝜑𝑀 <<s 𝑆)
7 ssltex1 27846 . . . . . . 7 (𝑀 <<s 𝑆𝑀 ∈ V)
86, 7syl 17 . . . . . 6 (𝜑𝑀 ∈ V)
91mpoexg 8100 . . . . . 6 ((𝐿 ∈ V ∧ 𝑀 ∈ V) → (𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V)
105, 8, 9syl2anc 584 . . . . 5 (𝜑 → (𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V)
11 rnexg 7925 . . . . 5 ((𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V → ran (𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V)
1210, 11syl 17 . . . 4 (𝜑 → ran (𝑝𝐿, 𝑞𝑀 ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V)
132, 12eqeltrrid 2844 . . 3 (𝜑 → {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∈ V)
14 eqid 2735 . . . . 5 (𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) = (𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
1514rnmpo 7566 . . . 4 ran (𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) = {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}
16 ssltex2 27847 . . . . . . 7 (𝐿 <<s 𝑅𝑅 ∈ V)
173, 16syl 17 . . . . . 6 (𝜑𝑅 ∈ V)
18 ssltex2 27847 . . . . . . 7 (𝑀 <<s 𝑆𝑆 ∈ V)
196, 18syl 17 . . . . . 6 (𝜑𝑆 ∈ V)
2014mpoexg 8100 . . . . . 6 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V)
2117, 19, 20syl2anc 584 . . . . 5 (𝜑 → (𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V)
22 rnexg 7925 . . . . 5 ((𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V → ran (𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V)
2321, 22syl 17 . . . 4 (𝜑 → ran (𝑟𝑅, 𝑠𝑆 ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V)
2415, 23eqeltrrid 2844 . . 3 (𝜑 → {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ∈ V)
2513, 24unexd 7773 . 2 (𝜑 → ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∈ V)
26 snex 5442 . . 3 {(𝐴 ·s 𝐵)} ∈ V
2726a1i 11 . 2 (𝜑 → {(𝐴 ·s 𝐵)} ∈ V)
28 ssltss1 27848 . . . . . . . . . . . 12 (𝐿 <<s 𝑅𝐿 No )
293, 28syl 17 . . . . . . . . . . 11 (𝜑𝐿 No )
3029adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐿 No )
31 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑝𝐿)
3230, 31sseldd 3996 . . . . . . . . 9 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑝 No )
33 ssltmul1.4 . . . . . . . . . . 11 (𝜑𝐵 = (𝑀 |s 𝑆))
346scutcld 27863 . . . . . . . . . . 11 (𝜑 → (𝑀 |s 𝑆) ∈ No )
3533, 34eqeltrd 2839 . . . . . . . . . 10 (𝜑𝐵 No )
3635adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐵 No )
3732, 36mulscld 28176 . . . . . . . 8 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (𝑝 ·s 𝐵) ∈ No )
38 ssltmul1.3 . . . . . . . . . . 11 (𝜑𝐴 = (𝐿 |s 𝑅))
393scutcld 27863 . . . . . . . . . . 11 (𝜑 → (𝐿 |s 𝑅) ∈ No )
4038, 39eqeltrd 2839 . . . . . . . . . 10 (𝜑𝐴 No )
4140adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐴 No )
42 ssltss1 27848 . . . . . . . . . . . 12 (𝑀 <<s 𝑆𝑀 No )
436, 42syl 17 . . . . . . . . . . 11 (𝜑𝑀 No )
4443adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑀 No )
45 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑞𝑀)
4644, 45sseldd 3996 . . . . . . . . 9 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑞 No )
4741, 46mulscld 28176 . . . . . . . 8 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (𝐴 ·s 𝑞) ∈ No )
4837, 47addscld 28028 . . . . . . 7 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) ∈ No )
4932, 46mulscld 28176 . . . . . . 7 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (𝑝 ·s 𝑞) ∈ No )
5048, 49subscld 28108 . . . . . 6 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No )
51 eleq1 2827 . . . . . 6 (𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (𝑎 No ↔ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No ))
5250, 51syl5ibrcom 247 . . . . 5 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑎 No ))
5352rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑎 No ))
5453abssdv 4078 . . 3 (𝜑 → {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ⊆ No )
55 ssltss2 27849 . . . . . . . . . . . 12 (𝐿 <<s 𝑅𝑅 No )
563, 55syl 17 . . . . . . . . . . 11 (𝜑𝑅 No )
5756adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑅 No )
58 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑟𝑅)
5957, 58sseldd 3996 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑟 No )
6035adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐵 No )
6159, 60mulscld 28176 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 ·s 𝐵) ∈ No )
6240adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 No )
63 ssltss2 27849 . . . . . . . . . . . 12 (𝑀 <<s 𝑆𝑆 No )
646, 63syl 17 . . . . . . . . . . 11 (𝜑𝑆 No )
6564adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑆 No )
66 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑠𝑆)
6765, 66sseldd 3996 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑠 No )
6862, 67mulscld 28176 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (𝐴 ·s 𝑠) ∈ No )
6961, 68addscld 28028 . . . . . . 7 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) ∈ No )
7059, 67mulscld 28176 . . . . . . 7 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 ·s 𝑠) ∈ No )
7169, 70subscld 28108 . . . . . 6 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No )
72 eleq1 2827 . . . . . 6 (𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (𝑏 No ↔ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No ))
7371, 72syl5ibrcom 247 . . . . 5 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → 𝑏 No ))
7473rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → 𝑏 No ))
7574abssdv 4078 . . 3 (𝜑 → {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ⊆ No )
7654, 75unssd 4202 . 2 (𝜑 → ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ⊆ No )
7740, 35mulscld 28176 . . 3 (𝜑 → (𝐴 ·s 𝐵) ∈ No )
7877snssd 4814 . 2 (𝜑 → {(𝐴 ·s 𝐵)} ⊆ No )
79 elun 4163 . . . . . . 7 (𝑥 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ↔ (𝑥 ∈ {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∨ 𝑥 ∈ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}))
80 vex 3482 . . . . . . . . 9 𝑥 ∈ V
81 eqeq1 2739 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
82812rexbidv 3220 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑝𝐿𝑞𝑀 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
8380, 82elab 3681 . . . . . . . 8 (𝑥 ∈ {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ↔ ∃𝑝𝐿𝑞𝑀 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
84 eqeq1 2739 . . . . . . . . . 10 (𝑏 = 𝑥 → (𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
85842rexbidv 3220 . . . . . . . . 9 (𝑏 = 𝑥 → (∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
8680, 85elab 3681 . . . . . . . 8 (𝑥 ∈ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
8783, 86orbi12i 914 . . . . . . 7 ((𝑥 ∈ {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∨ 𝑥 ∈ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ↔ (∃𝑝𝐿𝑞𝑀 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∨ ∃𝑟𝑅𝑠𝑆 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
8879, 87bitri 275 . . . . . 6 (𝑥 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ↔ (∃𝑝𝐿𝑞𝑀 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∨ ∃𝑟𝑅𝑠𝑆 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
8937, 47, 49addsubsd 28127 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑞)) +s (𝐴 ·s 𝑞)))
90 scutcut 27861 . . . . . . . . . . . . . . . 16 (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
913, 90syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
9291simp2d 1142 . . . . . . . . . . . . . 14 (𝜑𝐿 <<s {(𝐿 |s 𝑅)})
9392adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐿 <<s {(𝐿 |s 𝑅)})
94 ovex 7464 . . . . . . . . . . . . . . . 16 (𝐿 |s 𝑅) ∈ V
9594snid 4667 . . . . . . . . . . . . . . 15 (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}
9638, 95eqeltrdi 2847 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ {(𝐿 |s 𝑅)})
9796adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
9893, 31, 97ssltsepcd 27854 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑝 <s 𝐴)
99 scutcut 27861 . . . . . . . . . . . . . . . 16 (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈ No 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
1006, 99syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 |s 𝑆) ∈ No 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
101100simp2d 1142 . . . . . . . . . . . . . 14 (𝜑𝑀 <<s {(𝑀 |s 𝑆)})
102101adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑀 <<s {(𝑀 |s 𝑆)})
103 ovex 7464 . . . . . . . . . . . . . . . 16 (𝑀 |s 𝑆) ∈ V
104103snid 4667 . . . . . . . . . . . . . . 15 (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}
10533, 104eqeltrdi 2847 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ {(𝑀 |s 𝑆)})
106105adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
107102, 45, 106ssltsepcd 27854 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑞 <s 𝐵)
10832, 41, 46, 36, 98, 107sltmuld 28178 . . . . . . . . . . 11 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑞)) <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑞)))
10937, 49subscld 28108 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑞)) ∈ No )
11077adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (𝐴 ·s 𝐵) ∈ No )
111109, 47, 110sltaddsubd 28136 . . . . . . . . . . 11 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → ((((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑞)) +s (𝐴 ·s 𝑞)) <s (𝐴 ·s 𝐵) ↔ ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑞)) <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑞))))
112108, 111mpbird 257 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑞)) +s (𝐴 ·s 𝑞)) <s (𝐴 ·s 𝐵))
11389, 112eqbrtrd 5170 . . . . . . . . 9 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s (𝐴 ·s 𝐵))
114 breq1 5151 . . . . . . . . 9 (𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (𝑥 <s (𝐴 ·s 𝐵) ↔ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s (𝐴 ·s 𝐵)))
115113, 114syl5ibrcom 247 . . . . . . . 8 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → (𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑥 <s (𝐴 ·s 𝐵)))
116115rexlimdvva 3211 . . . . . . 7 (𝜑 → (∃𝑝𝐿𝑞𝑀 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑥 <s (𝐴 ·s 𝐵)))
11761, 68, 70addsubsd 28127 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) = (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑠)) +s (𝐴 ·s 𝑠)))
1183adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐿 <<s 𝑅)
119118, 90syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
120119simp3d 1143 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → {(𝐿 |s 𝑅)} <<s 𝑅)
12138adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 = (𝐿 |s 𝑅))
122121, 95eqeltrdi 2847 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
123120, 122, 58ssltsepcd 27854 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 <s 𝑟)
1246adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑀 <<s 𝑆)
125124, 99syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → ((𝑀 |s 𝑆) ∈ No 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
126125simp3d 1143 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → {(𝑀 |s 𝑆)} <<s 𝑆)
12733adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐵 = (𝑀 |s 𝑆))
128127, 104eqeltrdi 2847 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
129126, 128, 66ssltsepcd 27854 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐵 <s 𝑠)
13062, 59, 60, 67, 123, 129sltmuld 28178 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → ((𝐴 ·s 𝑠) -s (𝐴 ·s 𝐵)) <s ((𝑟 ·s 𝑠) -s (𝑟 ·s 𝐵)))
13161, 70subscld 28108 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → ((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑠)) ∈ No )
13277adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (𝐴 ·s 𝐵) ∈ No )
133131, 68, 132sltaddsubd 28136 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → ((((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑠)) +s (𝐴 ·s 𝑠)) <s (𝐴 ·s 𝐵) ↔ ((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑠)) <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑠))))
13461, 70, 132, 68sltsubsub2bd 28129 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑠)) <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑠)) ↔ ((𝐴 ·s 𝑠) -s (𝐴 ·s 𝐵)) <s ((𝑟 ·s 𝑠) -s (𝑟 ·s 𝐵))))
135133, 134bitrd 279 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → ((((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑠)) +s (𝐴 ·s 𝑠)) <s (𝐴 ·s 𝐵) ↔ ((𝐴 ·s 𝑠) -s (𝐴 ·s 𝐵)) <s ((𝑟 ·s 𝑠) -s (𝑟 ·s 𝐵))))
136130, 135mpbird 257 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑠)) +s (𝐴 ·s 𝑠)) <s (𝐴 ·s 𝐵))
137117, 136eqbrtrd 5170 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s (𝐴 ·s 𝐵))
138 breq1 5151 . . . . . . . . 9 (𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (𝑥 <s (𝐴 ·s 𝐵) ↔ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s (𝐴 ·s 𝐵)))
139137, 138syl5ibrcom 247 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → 𝑥 <s (𝐴 ·s 𝐵)))
140139rexlimdvva 3211 . . . . . . 7 (𝜑 → (∃𝑟𝑅𝑠𝑆 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → 𝑥 <s (𝐴 ·s 𝐵)))
141116, 140jaod 859 . . . . . 6 (𝜑 → ((∃𝑝𝐿𝑞𝑀 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∨ ∃𝑟𝑅𝑠𝑆 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) → 𝑥 <s (𝐴 ·s 𝐵)))
14288, 141biimtrid 242 . . . . 5 (𝜑 → (𝑥 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) → 𝑥 <s (𝐴 ·s 𝐵)))
143142imp 406 . . . 4 ((𝜑𝑥 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})) → 𝑥 <s (𝐴 ·s 𝐵))
144 velsn 4647 . . . . 5 (𝑦 ∈ {(𝐴 ·s 𝐵)} ↔ 𝑦 = (𝐴 ·s 𝐵))
145 breq2 5152 . . . . 5 (𝑦 = (𝐴 ·s 𝐵) → (𝑥 <s 𝑦𝑥 <s (𝐴 ·s 𝐵)))
146144, 145sylbi 217 . . . 4 (𝑦 ∈ {(𝐴 ·s 𝐵)} → (𝑥 <s 𝑦𝑥 <s (𝐴 ·s 𝐵)))
147143, 146syl5ibrcom 247 . . 3 ((𝜑𝑥 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})) → (𝑦 ∈ {(𝐴 ·s 𝐵)} → 𝑥 <s 𝑦))
1481473impia 1116 . 2 ((𝜑𝑥 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∧ 𝑦 ∈ {(𝐴 ·s 𝐵)}) → 𝑥 <s 𝑦)
14925, 27, 76, 78, 148ssltd 27851 1 (𝜑 → ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  cun 3961  wss 3963  {csn 4631   class class class wbr 5148  ran crn 5690  (class class class)co 7431  cmpo 7433   No csur 27699   <s cslt 27700   <<s csslt 27840   |s cscut 27842   +s cadds 28007   -s csubs 28067   ·s cmuls 28147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148
This theorem is referenced by:  mulsuniflem  28190
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