| Step | Hyp | Ref
| Expression |
| 1 | | oveq2 7439 |
. . . . . 6
⊢ (𝑣 = (◡𝐺‘ 0 ) → (𝑋 + 𝑣) = (𝑋 + (◡𝐺‘ 0 ))) |
| 2 | 1 | eqeq1d 2739 |
. . . . 5
⊢ (𝑣 = (◡𝐺‘ 0 ) → ((𝑋 + 𝑣) = 0 ↔ (𝑋 + (◡𝐺‘ 0 )) = 0 )) |
| 3 | | f1ocnv 6860 |
. . . . . . . 8
⊢ (𝐺:𝐵–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐵) |
| 4 | | f1of 6848 |
. . . . . . . 8
⊢ (◡𝐺:𝐵–1-1-onto→𝐵 → ◡𝐺:𝐵⟶𝐵) |
| 5 | 3, 4 | syl 17 |
. . . . . . 7
⊢ (𝐺:𝐵–1-1-onto→𝐵 → ◡𝐺:𝐵⟶𝐵) |
| 6 | 5 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ◡𝐺:𝐵⟶𝐵) |
| 7 | | mndractf1o.e |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ Mnd) |
| 8 | | mndractf1o.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝐸) |
| 9 | | mndractf1o.z |
. . . . . . . . 9
⊢ 0 =
(0g‘𝐸) |
| 10 | 8, 9 | mndidcl 18762 |
. . . . . . . 8
⊢ (𝐸 ∈ Mnd → 0 ∈ 𝐵) |
| 11 | 7, 10 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ 𝐵) |
| 12 | 11 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → 0 ∈ 𝐵) |
| 13 | 6, 12 | ffvelcdmd 7105 |
. . . . 5
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (◡𝐺‘ 0 ) ∈ 𝐵) |
| 14 | | f1of1 6847 |
. . . . . . 7
⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵–1-1→𝐵) |
| 15 | 14 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → 𝐺:𝐵–1-1→𝐵) |
| 16 | | mndractf1o.p |
. . . . . . . 8
⊢ + =
(+g‘𝐸) |
| 17 | 7 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → 𝐸 ∈ Mnd) |
| 18 | | mndractf1o.x |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 19 | 18 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → 𝑋 ∈ 𝐵) |
| 20 | 8, 16, 17, 19, 13 | mndcld 33027 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝑋 + (◡𝐺‘ 0 )) ∈ 𝐵) |
| 21 | 20, 12 | jca 511 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ((𝑋 + (◡𝐺‘ 0 )) ∈ 𝐵 ∧ 0 ∈ 𝐵)) |
| 22 | 8, 16, 9 | mndlid 18767 |
. . . . . . . 8
⊢ ((𝐸 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
| 23 | 17, 19, 22 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ( 0 + 𝑋) = 𝑋) |
| 24 | | mndractf1o.f |
. . . . . . . 8
⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋)) |
| 25 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑎 = 0 → (𝑎 + 𝑋) = ( 0 + 𝑋)) |
| 26 | | ovexd 7466 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ( 0 + 𝑋) ∈ V) |
| 27 | 24, 25, 12, 26 | fvmptd3 7039 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝐺‘ 0 ) = ( 0 + 𝑋)) |
| 28 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑎 = (𝑋 + (◡𝐺‘ 0 )) → (𝑎 + 𝑋) = ((𝑋 + (◡𝐺‘ 0 )) + 𝑋)) |
| 29 | | ovexd 7466 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ((𝑋 + (◡𝐺‘ 0 )) + 𝑋) ∈ V) |
| 30 | 24, 28, 20, 29 | fvmptd3 7039 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝐺‘(𝑋 + (◡𝐺‘ 0 ))) = ((𝑋 + (◡𝐺‘ 0 )) + 𝑋)) |
| 31 | 8, 16, 17, 19, 13, 19 | mndassd 33028 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ((𝑋 + (◡𝐺‘ 0 )) + 𝑋) = (𝑋 + ((◡𝐺‘ 0 ) + 𝑋))) |
| 32 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑎 = (◡𝐺‘ 0 ) → (𝑎 + 𝑋) = ((◡𝐺‘ 0 ) + 𝑋)) |
| 33 | | ovexd 7466 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ((◡𝐺‘ 0 ) + 𝑋) ∈ V) |
| 34 | 24, 32, 13, 33 | fvmptd3 7039 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝐺‘(◡𝐺‘ 0 )) = ((◡𝐺‘ 0 ) + 𝑋)) |
| 35 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → 𝐺:𝐵–1-1-onto→𝐵) |
| 36 | | f1ocnvfv2 7297 |
. . . . . . . . . . . 12
⊢ ((𝐺:𝐵–1-1-onto→𝐵 ∧ 0 ∈ 𝐵) → (𝐺‘(◡𝐺‘ 0 )) = 0 ) |
| 37 | 35, 12, 36 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝐺‘(◡𝐺‘ 0 )) = 0 ) |
| 38 | 34, 37 | eqtr3d 2779 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ((◡𝐺‘ 0 ) + 𝑋) = 0 ) |
| 39 | 38 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝑋 + ((◡𝐺‘ 0 ) + 𝑋)) = (𝑋 + 0 )) |
| 40 | 8, 16, 9 | mndrid 18768 |
. . . . . . . . . 10
⊢ ((𝐸 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
| 41 | 17, 19, 40 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝑋 + 0 ) = 𝑋) |
| 42 | 31, 39, 41 | 3eqtrd 2781 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ((𝑋 + (◡𝐺‘ 0 )) + 𝑋) = 𝑋) |
| 43 | 30, 42 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝐺‘(𝑋 + (◡𝐺‘ 0 ))) = 𝑋) |
| 44 | 23, 27, 43 | 3eqtr4rd 2788 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝐺‘(𝑋 + (◡𝐺‘ 0 ))) = (𝐺‘ 0 )) |
| 45 | | f1fveq 7282 |
. . . . . . 7
⊢ ((𝐺:𝐵–1-1→𝐵 ∧ ((𝑋 + (◡𝐺‘ 0 )) ∈ 𝐵 ∧ 0 ∈ 𝐵)) → ((𝐺‘(𝑋 + (◡𝐺‘ 0 ))) = (𝐺‘ 0 ) ↔ (𝑋 + (◡𝐺‘ 0 )) = 0 )) |
| 46 | 45 | biimpa 476 |
. . . . . 6
⊢ (((𝐺:𝐵–1-1→𝐵 ∧ ((𝑋 + (◡𝐺‘ 0 )) ∈ 𝐵 ∧ 0 ∈ 𝐵)) ∧ (𝐺‘(𝑋 + (◡𝐺‘ 0 ))) = (𝐺‘ 0 )) → (𝑋 + (◡𝐺‘ 0 )) = 0 ) |
| 47 | 15, 21, 44, 46 | syl21anc 838 |
. . . . 5
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (𝑋 + (◡𝐺‘ 0 )) = 0 ) |
| 48 | 2, 13, 47 | rspcedvdw 3625 |
. . . 4
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ∃𝑣 ∈ 𝐵 (𝑋 + 𝑣) = 0 ) |
| 49 | | f1ofo 6855 |
. . . . 5
⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵–onto→𝐵) |
| 50 | 8, 9, 16, 24, 7, 18 | mndractfo 33034 |
. . . . . 6
⊢ (𝜑 → (𝐺:𝐵–onto→𝐵 ↔ ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 )) |
| 51 | 50 | biimpa 476 |
. . . . 5
⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 ) |
| 52 | 49, 51 | sylan2 593 |
. . . 4
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 ) |
| 53 | 48, 52 | jca 511 |
. . 3
⊢ ((𝜑 ∧ 𝐺:𝐵–1-1-onto→𝐵) → (∃𝑣 ∈ 𝐵 (𝑋 + 𝑣) = 0 ∧ ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 )) |
| 54 | 7 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑋 + 𝑣) = 0 ) → 𝐸 ∈ Mnd) |
| 55 | 18 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑋 + 𝑣) = 0 ) → 𝑋 ∈ 𝐵) |
| 56 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑋 + 𝑣) = 0 ) → 𝑣 ∈ 𝐵) |
| 57 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑋 + 𝑣) = 0 ) → (𝑋 + 𝑣) = 0 ) |
| 58 | 8, 9, 16, 24, 54, 55, 56, 57 | mndractf1 33033 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑋 + 𝑣) = 0 ) → 𝐺:𝐵–1-1→𝐵) |
| 59 | 58 | r19.29an 3158 |
. . . . 5
⊢ ((𝜑 ∧ ∃𝑣 ∈ 𝐵 (𝑋 + 𝑣) = 0 ) → 𝐺:𝐵–1-1→𝐵) |
| 60 | 50 | biimpar 477 |
. . . . 5
⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 ) → 𝐺:𝐵–onto→𝐵) |
| 61 | 59, 60 | anim12dan 619 |
. . . 4
⊢ ((𝜑 ∧ (∃𝑣 ∈ 𝐵 (𝑋 + 𝑣) = 0 ∧ ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 )) → (𝐺:𝐵–1-1→𝐵 ∧ 𝐺:𝐵–onto→𝐵)) |
| 62 | | df-f1o 6568 |
. . . 4
⊢ (𝐺:𝐵–1-1-onto→𝐵 ↔ (𝐺:𝐵–1-1→𝐵 ∧ 𝐺:𝐵–onto→𝐵)) |
| 63 | 61, 62 | sylibr 234 |
. . 3
⊢ ((𝜑 ∧ (∃𝑣 ∈ 𝐵 (𝑋 + 𝑣) = 0 ∧ ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 )) → 𝐺:𝐵–1-1-onto→𝐵) |
| 64 | 53, 63 | impbida 801 |
. 2
⊢ (𝜑 → (𝐺:𝐵–1-1-onto→𝐵 ↔ (∃𝑣 ∈ 𝐵 (𝑋 + 𝑣) = 0 ∧ ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 ))) |
| 65 | 8, 9, 16, 7, 18 | mndlrinvb 33030 |
. 2
⊢ (𝜑 → ((∃𝑣 ∈ 𝐵 (𝑋 + 𝑣) = 0 ∧ ∃𝑤 ∈ 𝐵 (𝑤 + 𝑋) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))) |
| 66 | 64, 65 | bitrd 279 |
1
⊢ (𝜑 → (𝐺:𝐵–1-1-onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))) |