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Theorem addsdilem4 27538
Description: Lemma for addsdi 27539. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)
Hypotheses
Ref Expression
addsdilem4.1 (𝜑𝐴 No )
addsdilem4.2 (𝜑𝐵 No )
addsdilem4.3 (𝜑𝐶 No )
addsdilem4.4 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
addsdilem4.5 (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
addsdilem4.6 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
addsdilem4.7 (𝜓𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
addsdilem4.8 (𝜓𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
Assertion
Ref Expression
addsdilem4 ((𝜑𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
Distinct variable groups:   𝐴,𝑥𝑂,𝑧𝑂   𝐵,𝑥𝑂,𝑧𝑂   𝐶,𝑥𝑂,𝑧𝑂   𝑋,𝑥𝑂,𝑧𝑂   𝑍,𝑧𝑂
Allowed substitution hints:   𝜑(𝑥𝑂,𝑧𝑂)   𝜓(𝑥𝑂,𝑧𝑂)   𝑍(𝑥𝑂)

Proof of Theorem addsdilem4
StepHypRef Expression
1 oveq1 7401 . . . . . 6 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2 oveq1 7401 . . . . . . 7 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3 oveq1 7401 . . . . . . 7 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
42, 3oveq12d 7412 . . . . . 6 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
51, 4eqeq12d 2748 . . . . 5 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) ↔ (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶))))
6 addsdilem4.4 . . . . . 6 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
76adantr 481 . . . . 5 ((𝜑𝜓) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
8 addsdilem4.7 . . . . . 6 (𝜓𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
98adantl 482 . . . . 5 ((𝜑𝜓) → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
105, 7, 9rspcdva 3611 . . . 4 ((𝜑𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11 oveq2 7402 . . . . . . 7 (𝑧𝑂 = 𝑍 → (𝐵 +s 𝑧𝑂) = (𝐵 +s 𝑍))
1211oveq2d 7410 . . . . . 6 (𝑧𝑂 = 𝑍 → (𝐴 ·s (𝐵 +s 𝑧𝑂)) = (𝐴 ·s (𝐵 +s 𝑍)))
13 oveq2 7402 . . . . . . 7 (𝑧𝑂 = 𝑍 → (𝐴 ·s 𝑧𝑂) = (𝐴 ·s 𝑍))
1413oveq2d 7410 . . . . . 6 (𝑧𝑂 = 𝑍 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
1512, 14eqeq12d 2748 . . . . 5 (𝑧𝑂 = 𝑍 → ((𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) ↔ (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
16 addsdilem4.5 . . . . . 6 (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
1716adantr 481 . . . . 5 ((𝜑𝜓) → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
18 addsdilem4.8 . . . . . 6 (𝜓𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
1918adantl 482 . . . . 5 ((𝜑𝜓) → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
2015, 17, 19rspcdva 3611 . . . 4 ((𝜑𝜓) → (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
2110, 20oveq12d 7412 . . 3 ((𝜑𝜓) → ((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) = (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
22 oveq1 7401 . . . . 5 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑧𝑂)))
23 oveq1 7401 . . . . . 6 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑧𝑂) = (𝑋 ·s 𝑧𝑂))
242, 23oveq12d 7412 . . . . 5 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)))
2522, 24eqeq12d 2748 . . . 4 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) ↔ (𝑋 ·s (𝐵 +s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂))))
2611oveq2d 7410 . . . . 5 (𝑧𝑂 = 𝑍 → (𝑋 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑍)))
27 oveq2 7402 . . . . . 6 (𝑧𝑂 = 𝑍 → (𝑋 ·s 𝑧𝑂) = (𝑋 ·s 𝑍))
2827oveq2d 7410 . . . . 5 (𝑧𝑂 = 𝑍 → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
2926, 28eqeq12d 2748 . . . 4 (𝑧𝑂 = 𝑍 → ((𝑋 ·s (𝐵 +s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) ↔ (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
30 addsdilem4.6 . . . . 5 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
3130adantr 481 . . . 4 ((𝜑𝜓) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
3225, 29, 31, 9, 19rspc2dv 3623 . . 3 ((𝜑𝜓) → (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
3321, 32oveq12d 7412 . 2 ((𝜑𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
34 leftssno 27304 . . . . . . . . 9 ( L ‘𝐴) ⊆ No
35 rightssno 27305 . . . . . . . . 9 ( R ‘𝐴) ⊆ No
3634, 35unssi 4182 . . . . . . . 8 (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
3736, 8sselid 3977 . . . . . . 7 (𝜓𝑋 No )
3837adantl 482 . . . . . 6 ((𝜑𝜓) → 𝑋 No )
39 addsdilem4.2 . . . . . . 7 (𝜑𝐵 No )
4039adantr 481 . . . . . 6 ((𝜑𝜓) → 𝐵 No )
4138, 40mulscld 27520 . . . . 5 ((𝜑𝜓) → (𝑋 ·s 𝐵) ∈ No )
42 addsdilem4.3 . . . . . . 7 (𝜑𝐶 No )
4342adantr 481 . . . . . 6 ((𝜑𝜓) → 𝐶 No )
4438, 43mulscld 27520 . . . . 5 ((𝜑𝜓) → (𝑋 ·s 𝐶) ∈ No )
4541, 44addscld 27393 . . . 4 ((𝜑𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) ∈ No )
46 addsdilem4.1 . . . . . . 7 (𝜑𝐴 No )
4746, 39mulscld 27520 . . . . . 6 (𝜑 → (𝐴 ·s 𝐵) ∈ No )
4847adantr 481 . . . . 5 ((𝜑𝜓) → (𝐴 ·s 𝐵) ∈ No )
4946adantr 481 . . . . . 6 ((𝜑𝜓) → 𝐴 No )
50 leftssno 27304 . . . . . . . . 9 ( L ‘𝐶) ⊆ No
51 rightssno 27305 . . . . . . . . 9 ( R ‘𝐶) ⊆ No
5250, 51unssi 4182 . . . . . . . 8 (( L ‘𝐶) ∪ ( R ‘𝐶)) ⊆ No
5352, 18sselid 3977 . . . . . . 7 (𝜓𝑍 No )
5453adantl 482 . . . . . 6 ((𝜑𝜓) → 𝑍 No )
5549, 54mulscld 27520 . . . . 5 ((𝜑𝜓) → (𝐴 ·s 𝑍) ∈ No )
5648, 55addscld 27393 . . . 4 ((𝜑𝜓) → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)) ∈ No )
5745, 56addscld 27393 . . 3 ((𝜑𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) ∈ No )
5838, 54mulscld 27520 . . 3 ((𝜑𝜓) → (𝑋 ·s 𝑍) ∈ No )
5957, 41, 58subsubs4d 27489 . 2 ((𝜑𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
6045, 56, 41addsubsd 27478 . . . . 5 ((𝜑𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
6141, 44addscomd 27380 . . . . . . . 8 ((𝜑𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)))
6261oveq1d 7409 . . . . . . 7 ((𝜑𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)))
63 pncans 27469 . . . . . . . 8 (((𝑋 ·s 𝐶) ∈ No ∧ (𝑋 ·s 𝐵) ∈ No ) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
6444, 41, 63syl2anc 584 . . . . . . 7 ((𝜑𝜓) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
6562, 64eqtrd 2772 . . . . . 6 ((𝜑𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
6665oveq1d 7409 . . . . 5 ((𝜑𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) = ((𝑋 ·s 𝐶) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
6744, 48, 55adds12d 27420 . . . . 5 ((𝜑𝜓) → ((𝑋 ·s 𝐶) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) = ((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))))
6860, 66, 673eqtrd 2776 . . . 4 ((𝜑𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) = ((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))))
6968oveq1d 7409 . . 3 ((𝜑𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = (((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝑍)))
7044, 55addscld 27393 . . . 4 ((𝜑𝜓) → ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) ∈ No )
7148, 70, 58addsubsassd 27477 . . 3 ((𝜑𝜓) → (((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝑍)) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
7269, 71eqtrd 2772 . 2 ((𝜑𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
7333, 59, 723eqtr2d 2778 1 ((𝜑𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  cun 3943  cfv 6533  (class class class)co 7394   No csur 27072   L cleft 27269   R cright 27270   +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:  addsdi  27539
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