| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mulsunif2.1 | . . 3
⊢ (𝜑 → 𝐿 <<s 𝑅) | 
| 2 |  | mulsunif2.2 | . . 3
⊢ (𝜑 → 𝑀 <<s 𝑆) | 
| 3 |  | mulsunif2.3 | . . 3
⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) | 
| 4 |  | mulsunif2.4 | . . 3
⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆)) | 
| 5 | 1, 2, 3, 4 | mulsunif2lem 28195 | . 2
⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) |s ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))}))) | 
| 6 |  | eqeq1 2741 | . . . . . . 7
⊢ (𝑒 = 𝑎 → (𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))))) | 
| 7 | 6 | 2rexbidv 3222 | . . . . . 6
⊢ (𝑒 = 𝑎 → (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))))) | 
| 8 |  | oveq2 7439 | . . . . . . . . . 10
⊢ (𝑖 = 𝑝 → (𝐴 -s 𝑖) = (𝐴 -s 𝑝)) | 
| 9 | 8 | oveq1d 7446 | . . . . . . . . 9
⊢ (𝑖 = 𝑝 → ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))) | 
| 10 | 9 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑖 = 𝑝 → ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)))) | 
| 11 | 10 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑖 = 𝑝 → (𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))))) | 
| 12 |  | oveq2 7439 | . . . . . . . . . 10
⊢ (𝑗 = 𝑞 → (𝐵 -s 𝑗) = (𝐵 -s 𝑞)) | 
| 13 | 12 | oveq2d 7447 | . . . . . . . . 9
⊢ (𝑗 = 𝑞 → ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞))) | 
| 14 | 13 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑗 = 𝑞 → ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))) | 
| 15 | 14 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑗 = 𝑞 → (𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞))))) | 
| 16 | 11, 15 | cbvrex2vw 3242 | . . . . . 6
⊢
(∃𝑖 ∈
𝐿 ∃𝑗 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))) | 
| 17 | 7, 16 | bitrdi 287 | . . . . 5
⊢ (𝑒 = 𝑎 → (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞))))) | 
| 18 | 17 | cbvabv 2812 | . . . 4
⊢ {𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} = {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} | 
| 19 |  | eqeq1 2741 | . . . . . . 7
⊢ (𝑓 = 𝑏 → (𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))))) | 
| 20 | 19 | 2rexbidv 3222 | . . . . . 6
⊢ (𝑓 = 𝑏 → (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))))) | 
| 21 |  | oveq1 7438 | . . . . . . . . . 10
⊢ (𝑘 = 𝑟 → (𝑘 -s 𝐴) = (𝑟 -s 𝐴)) | 
| 22 | 21 | oveq1d 7446 | . . . . . . . . 9
⊢ (𝑘 = 𝑟 → ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))) | 
| 23 | 22 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑘 = 𝑟 → ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)))) | 
| 24 | 23 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑘 = 𝑟 → (𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))))) | 
| 25 |  | oveq1 7438 | . . . . . . . . . 10
⊢ (𝑙 = 𝑠 → (𝑙 -s 𝐵) = (𝑠 -s 𝐵)) | 
| 26 | 25 | oveq2d 7447 | . . . . . . . . 9
⊢ (𝑙 = 𝑠 → ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵))) | 
| 27 | 26 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑙 = 𝑠 → ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))) = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))) | 
| 28 | 27 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑙 = 𝑠 → (𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵))))) | 
| 29 | 24, 28 | cbvrex2vw 3242 | . . . . . 6
⊢
(∃𝑘 ∈
𝑅 ∃𝑙 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))) | 
| 30 | 20, 29 | bitrdi 287 | . . . . 5
⊢ (𝑓 = 𝑏 → (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵))))) | 
| 31 | 30 | cbvabv 2812 | . . . 4
⊢ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))} = {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))} | 
| 32 | 18, 31 | uneq12i 4166 | . . 3
⊢ ({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) = ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) | 
| 33 |  | eqeq1 2741 | . . . . . . 7
⊢ (𝑔 = 𝑐 → (𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))))) | 
| 34 | 33 | 2rexbidv 3222 | . . . . . 6
⊢ (𝑔 = 𝑐 → (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))))) | 
| 35 |  | oveq2 7439 | . . . . . . . . . 10
⊢ (𝑚 = 𝑡 → (𝐴 -s 𝑚) = (𝐴 -s 𝑡)) | 
| 36 | 35 | oveq1d 7446 | . . . . . . . . 9
⊢ (𝑚 = 𝑡 → ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))) | 
| 37 | 36 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑚 = 𝑡 → ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)))) | 
| 38 | 37 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑚 = 𝑡 → (𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))))) | 
| 39 |  | oveq1 7438 | . . . . . . . . . 10
⊢ (𝑛 = 𝑢 → (𝑛 -s 𝐵) = (𝑢 -s 𝐵)) | 
| 40 | 39 | oveq2d 7447 | . . . . . . . . 9
⊢ (𝑛 = 𝑢 → ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))) | 
| 41 | 40 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑛 = 𝑢 → ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) | 
| 42 | 41 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑛 = 𝑢 → (𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))))) | 
| 43 | 38, 42 | cbvrex2vw 3242 | . . . . . 6
⊢
(∃𝑚 ∈
𝐿 ∃𝑛 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))) | 
| 44 | 34, 43 | bitrdi 287 | . . . . 5
⊢ (𝑔 = 𝑐 → (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))))) | 
| 45 | 44 | cbvabv 2812 | . . . 4
⊢ {𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} = {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} | 
| 46 |  | eqeq1 2741 | . . . . . . 7
⊢ (ℎ = 𝑑 → (ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))))) | 
| 47 | 46 | 2rexbidv 3222 | . . . . . 6
⊢ (ℎ = 𝑑 → (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))))) | 
| 48 |  | oveq1 7438 | . . . . . . . . . 10
⊢ (𝑜 = 𝑣 → (𝑜 -s 𝐴) = (𝑣 -s 𝐴)) | 
| 49 | 48 | oveq1d 7446 | . . . . . . . . 9
⊢ (𝑜 = 𝑣 → ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))) | 
| 50 | 49 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑜 = 𝑣 → ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)))) | 
| 51 | 50 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑜 = 𝑣 → (𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))))) | 
| 52 |  | oveq2 7439 | . . . . . . . . . 10
⊢ (𝑥 = 𝑤 → (𝐵 -s 𝑥) = (𝐵 -s 𝑤)) | 
| 53 | 52 | oveq2d 7447 | . . . . . . . . 9
⊢ (𝑥 = 𝑤 → ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤))) | 
| 54 | 53 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑥 = 𝑤 → ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))) = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))) | 
| 55 | 54 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑥 = 𝑤 → (𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤))))) | 
| 56 | 51, 55 | cbvrex2vw 3242 | . . . . . 6
⊢
(∃𝑜 ∈
𝑅 ∃𝑥 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))) | 
| 57 | 47, 56 | bitrdi 287 | . . . . 5
⊢ (ℎ = 𝑑 → (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤))))) | 
| 58 | 57 | cbvabv 2812 | . . . 4
⊢ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))} = {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))} | 
| 59 | 45, 58 | uneq12i 4166 | . . 3
⊢ ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))}) = ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))}) | 
| 60 | 32, 59 | oveq12i 7443 | . 2
⊢ (({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) |s ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))})) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))})) | 
| 61 | 5, 60 | eqtrdi 2793 | 1
⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))}))) |