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Theorem ssltmul2 27532
Description: One surreal set less-than relationship for cuts of 𝐴 and 𝐵. (Contributed by Scott Fenton, 7-Mar-2025.)
Hypotheses
Ref Expression
ssltmul2.1 (𝜑𝐿 <<s 𝑅)
ssltmul2.2 (𝜑𝑀 <<s 𝑆)
ssltmul2.3 (𝜑𝐴 = (𝐿 |s 𝑅))
ssltmul2.4 (𝜑𝐵 = (𝑀 |s 𝑆))
Assertion
Ref Expression
ssltmul2 (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
Distinct variable groups:   𝐴,𝑐   𝐴,𝑑   𝑡,𝐴,𝑢   𝑣,𝐴,𝑤   𝐵,𝑐   𝐵,𝑑   𝑡,𝐵,𝑢   𝑣,𝐵,𝑤   𝐿,𝑐,𝑡,𝑢   𝑀,𝑑,𝑣,𝑤   𝑅,𝑑,𝑣,𝑤   𝑆,𝑐,𝑡,𝑢   𝜑,𝑐,𝑡,𝑢   𝜑,𝑑,𝑣,𝑤
Allowed substitution hints:   𝑅(𝑢,𝑡,𝑐)   𝑆(𝑤,𝑣,𝑑)   𝐿(𝑤,𝑣,𝑑)   𝑀(𝑢,𝑡,𝑐)

Proof of Theorem ssltmul2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5425 . . 3 {(𝐴 ·s 𝐵)} ∈ V
21a1i 11 . 2 (𝜑 → {(𝐴 ·s 𝐵)} ∈ V)
3 eqid 2732 . . . . 5 (𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) = (𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
43rnmpo 7526 . . . 4 ran (𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) = {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))}
5 ssltmul2.1 . . . . . . 7 (𝜑𝐿 <<s 𝑅)
6 ssltex1 27217 . . . . . . 7 (𝐿 <<s 𝑅𝐿 ∈ V)
75, 6syl 17 . . . . . 6 (𝜑𝐿 ∈ V)
8 ssltmul2.2 . . . . . . 7 (𝜑𝑀 <<s 𝑆)
9 ssltex2 27218 . . . . . . 7 (𝑀 <<s 𝑆𝑆 ∈ V)
108, 9syl 17 . . . . . 6 (𝜑𝑆 ∈ V)
113mpoexg 8047 . . . . . 6 ((𝐿 ∈ V ∧ 𝑆 ∈ V) → (𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V)
127, 10, 11syl2anc 584 . . . . 5 (𝜑 → (𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V)
13 rnexg 7879 . . . . 5 ((𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V → ran (𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V)
1412, 13syl 17 . . . 4 (𝜑 → ran (𝑡𝐿, 𝑢𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V)
154, 14eqeltrrid 2838 . . 3 (𝜑 → {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∈ V)
16 eqid 2732 . . . . 5 (𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) = (𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
1716rnmpo 7526 . . . 4 ran (𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) = {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}
18 ssltex2 27218 . . . . . . 7 (𝐿 <<s 𝑅𝑅 ∈ V)
195, 18syl 17 . . . . . 6 (𝜑𝑅 ∈ V)
20 ssltex1 27217 . . . . . . 7 (𝑀 <<s 𝑆𝑀 ∈ V)
218, 20syl 17 . . . . . 6 (𝜑𝑀 ∈ V)
2216mpoexg 8047 . . . . . 6 ((𝑅 ∈ V ∧ 𝑀 ∈ V) → (𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V)
2319, 21, 22syl2anc 584 . . . . 5 (𝜑 → (𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V)
24 rnexg 7879 . . . . 5 ((𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V → ran (𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V)
2523, 24syl 17 . . . 4 (𝜑 → ran (𝑣𝑅, 𝑤𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V)
2617, 25eqeltrrid 2838 . . 3 (𝜑 → {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ∈ V)
2715, 26unexd 7725 . 2 (𝜑 → ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ∈ V)
28 ssltmul2.3 . . . . 5 (𝜑𝐴 = (𝐿 |s 𝑅))
295scutcld 27233 . . . . 5 (𝜑 → (𝐿 |s 𝑅) ∈ No )
3028, 29eqeltrd 2833 . . . 4 (𝜑𝐴 No )
31 ssltmul2.4 . . . . 5 (𝜑𝐵 = (𝑀 |s 𝑆))
328scutcld 27233 . . . . 5 (𝜑 → (𝑀 |s 𝑆) ∈ No )
3331, 32eqeltrd 2833 . . . 4 (𝜑𝐵 No )
3430, 33mulscld 27520 . . 3 (𝜑 → (𝐴 ·s 𝐵) ∈ No )
3534snssd 4806 . 2 (𝜑 → {(𝐴 ·s 𝐵)} ⊆ No )
36 ssltss1 27219 . . . . . . . . . . . 12 (𝐿 <<s 𝑅𝐿 No )
375, 36syl 17 . . . . . . . . . . 11 (𝜑𝐿 No )
3837adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐿 No )
39 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑡𝐿)
4038, 39sseldd 3980 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑡 No )
4133adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐵 No )
4240, 41mulscld 27520 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝑡 ·s 𝐵) ∈ No )
4330adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐴 No )
44 ssltss2 27220 . . . . . . . . . . . 12 (𝑀 <<s 𝑆𝑆 No )
458, 44syl 17 . . . . . . . . . . 11 (𝜑𝑆 No )
4645adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑆 No )
47 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑢𝑆)
4846, 47sseldd 3980 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑢 No )
4943, 48mulscld 27520 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝐴 ·s 𝑢) ∈ No )
5042, 49addscld 27393 . . . . . . 7 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
5140, 48mulscld 27520 . . . . . . 7 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝑡 ·s 𝑢) ∈ No )
5250, 51subscld 27464 . . . . . 6 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No )
53 eleq1 2821 . . . . . 6 (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (𝑐 No ↔ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No ))
5452, 53syl5ibrcom 246 . . . . 5 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑐 No ))
5554rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑐 No ))
5655abssdv 4062 . . 3 (𝜑 → {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ No )
57 ssltss2 27220 . . . . . . . . . . . 12 (𝐿 <<s 𝑅𝑅 No )
585, 57syl 17 . . . . . . . . . . 11 (𝜑𝑅 No )
5958adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑅 No )
60 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑣𝑅)
6159, 60sseldd 3980 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑣 No )
6233adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝐵 No )
6361, 62mulscld 27520 . . . . . . . 8 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝑣 ·s 𝐵) ∈ No )
6430adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝐴 No )
65 ssltss1 27219 . . . . . . . . . . . 12 (𝑀 <<s 𝑆𝑀 No )
668, 65syl 17 . . . . . . . . . . 11 (𝜑𝑀 No )
6766adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑀 No )
68 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑤𝑀)
6967, 68sseldd 3980 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑤 No )
7064, 69mulscld 27520 . . . . . . . 8 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝐴 ·s 𝑤) ∈ No )
7163, 70addscld 27393 . . . . . . 7 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No )
7261, 69mulscld 27520 . . . . . . 7 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝑣 ·s 𝑤) ∈ No )
7371, 72subscld 27464 . . . . . 6 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
74 eleq1 2821 . . . . . 6 (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (𝑑 No ↔ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No ))
7573, 74syl5ibrcom 246 . . . . 5 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑑 No ))
7675rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑑 No ))
7776abssdv 4062 . . 3 (𝜑 → {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ No )
7856, 77unssd 4183 . 2 (𝜑 → ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ⊆ No )
79 elun 4145 . . . . . 6 (𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ↔ (𝑦 ∈ {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∨ 𝑦 ∈ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
80 vex 3478 . . . . . . . 8 𝑦 ∈ V
81 eqeq1 2736 . . . . . . . . 9 (𝑐 = 𝑦 → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
82812rexbidv 3219 . . . . . . . 8 (𝑐 = 𝑦 → (∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑡𝐿𝑢𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
8380, 82elab 3665 . . . . . . 7 (𝑦 ∈ {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ↔ ∃𝑡𝐿𝑢𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
84 eqeq1 2736 . . . . . . . . 9 (𝑑 = 𝑦 → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
85842rexbidv 3219 . . . . . . . 8 (𝑑 = 𝑦 → (∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑣𝑅𝑤𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
8680, 85elab 3665 . . . . . . 7 (𝑦 ∈ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ↔ ∃𝑣𝑅𝑤𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
8783, 86orbi12i 913 . . . . . 6 ((𝑦 ∈ {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∨ 𝑦 ∈ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ↔ (∃𝑡𝐿𝑢𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∨ ∃𝑣𝑅𝑤𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
8879, 87bitri 274 . . . . 5 (𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ↔ (∃𝑡𝐿𝑢𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∨ ∃𝑣𝑅𝑤𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
89 scutcut 27231 . . . . . . . . . . . . . . 15 (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
905, 89syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
9190simp2d 1143 . . . . . . . . . . . . 13 (𝜑𝐿 <<s {(𝐿 |s 𝑅)})
9291adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐿 <<s {(𝐿 |s 𝑅)})
93 ovex 7427 . . . . . . . . . . . . . . 15 (𝐿 |s 𝑅) ∈ V
9493snid 4659 . . . . . . . . . . . . . 14 (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}
9528, 94eqeltrdi 2841 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ {(𝐿 |s 𝑅)})
9695adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
9792, 39, 96ssltsepcd 27224 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑡 <s 𝐴)
98 scutcut 27231 . . . . . . . . . . . . . . 15 (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈ No 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
998, 98syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 |s 𝑆) ∈ No 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
10099simp3d 1144 . . . . . . . . . . . . 13 (𝜑 → {(𝑀 |s 𝑆)} <<s 𝑆)
101100adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → {(𝑀 |s 𝑆)} <<s 𝑆)
102 ovex 7427 . . . . . . . . . . . . . . 15 (𝑀 |s 𝑆) ∈ V
103102snid 4659 . . . . . . . . . . . . . 14 (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}
10431, 103eqeltrdi 2841 . . . . . . . . . . . . 13 (𝜑𝐵 ∈ {(𝑀 |s 𝑆)})
105104adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
106101, 105, 47ssltsepcd 27224 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐵 <s 𝑢)
10740, 43, 41, 48, 97, 106sltmuld 27522 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) -s (𝐴 ·s 𝐵)))
10834adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝐴 ·s 𝐵) ∈ No )
10951, 42, 49, 108sltsubsub2bd 27480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) -s (𝐴 ·s 𝐵)) ↔ ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑢)) <s ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢))))
11042, 51subscld 27464 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ∈ No )
111108, 49, 110sltsubaddd 27485 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑢)) <s ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ↔ (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢))))
112109, 111bitrd 278 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) -s (𝐴 ·s 𝐵)) ↔ (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢))))
113107, 112mpbid 231 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
11442, 49, 51addsubsd 27478 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) = (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
115113, 114breqtrrd 5170 . . . . . . . 8 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
116 breq2 5146 . . . . . . . 8 (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → ((𝐴 ·s 𝐵) <s 𝑦 ↔ (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
117115, 116syl5ibrcom 246 . . . . . . 7 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (𝐴 ·s 𝐵) <s 𝑦))
118117rexlimdvva 3211 . . . . . 6 (𝜑 → (∃𝑡𝐿𝑢𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (𝐴 ·s 𝐵) <s 𝑦))
11990simp3d 1144 . . . . . . . . . . . . 13 (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
120119adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → {(𝐿 |s 𝑅)} <<s 𝑅)
12195adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
122120, 121, 60ssltsepcd 27224 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝐴 <s 𝑣)
12399simp2d 1143 . . . . . . . . . . . . 13 (𝜑𝑀 <<s {(𝑀 |s 𝑆)})
124123adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑀 <<s {(𝑀 |s 𝑆)})
125104adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
126124, 68, 125ssltsepcd 27224 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑤 <s 𝐵)
12764, 61, 69, 62, 122, 126sltmuld 27522 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑤)) <s ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)))
12834adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝐴 ·s 𝐵) ∈ No )
12963, 72subscld 27464 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) ∈ No )
130128, 70, 129sltsubaddd 27485 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑤)) <s ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) ↔ (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤))))
131127, 130mpbid 231 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
13263, 70, 72addsubsd 27478 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) = (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
133131, 132breqtrrd 5170 . . . . . . . 8 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
134 breq2 5146 . . . . . . . 8 (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → ((𝐴 ·s 𝐵) <s 𝑦 ↔ (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
135133, 134syl5ibrcom 246 . . . . . . 7 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (𝐴 ·s 𝐵) <s 𝑦))
136135rexlimdvva 3211 . . . . . 6 (𝜑 → (∃𝑣𝑅𝑤𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (𝐴 ·s 𝐵) <s 𝑦))
137118, 136jaod 857 . . . . 5 (𝜑 → ((∃𝑡𝐿𝑢𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∨ ∃𝑣𝑅𝑤𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) → (𝐴 ·s 𝐵) <s 𝑦))
13888, 137biimtrid 241 . . . 4 (𝜑 → (𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → (𝐴 ·s 𝐵) <s 𝑦))
139 velsn 4639 . . . . 5 (𝑥 ∈ {(𝐴 ·s 𝐵)} ↔ 𝑥 = (𝐴 ·s 𝐵))
140 breq1 5145 . . . . . 6 (𝑥 = (𝐴 ·s 𝐵) → (𝑥 <s 𝑦 ↔ (𝐴 ·s 𝐵) <s 𝑦))
141140imbi2d 340 . . . . 5 (𝑥 = (𝐴 ·s 𝐵) → ((𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → 𝑥 <s 𝑦) ↔ (𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → (𝐴 ·s 𝐵) <s 𝑦)))
142139, 141sylbi 216 . . . 4 (𝑥 ∈ {(𝐴 ·s 𝐵)} → ((𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → 𝑥 <s 𝑦) ↔ (𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → (𝐴 ·s 𝐵) <s 𝑦)))
143138, 142syl5ibrcom 246 . . 3 (𝜑 → (𝑥 ∈ {(𝐴 ·s 𝐵)} → (𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → 𝑥 <s 𝑦)))
1441433imp 1111 . 2 ((𝜑𝑥 ∈ {(𝐴 ·s 𝐵)} ∧ 𝑦 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) → 𝑥 <s 𝑦)
1452, 27, 35, 78, 144ssltd 27222 1 (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  {cab 2709  wrex 3070  Vcvv 3474  cun 3943  wss 3945  {csn 4623   class class class wbr 5142  ran crn 5671  (class class class)co 7394  cmpo 7396   No csur 27072   <s cslt 27073   <<s csslt 27211   |s cscut 27213   +s cadds 27372   -s csubs 27424   ·s cmuls 27491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492
This theorem is referenced by:  mulsuniflem  27533
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